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LIBRARY University of California. Class THE PRACTICE OF NAVIGATION AND NAUTICAL ASTRONOMY. THE PEAGTICE OF NAVIGATION NAUTICAL ASTEONOMY. HENRY RAPER, Lieut. R.N., F.R.A.S., F.R.G.S NINETEENTH EDITION. REVISED AND ENLARGED. Uonlfon : PUBLISHED BY J. D. POTTER, Admiralty Agent far Cliarls 145 MINORIES, E. 1908 ^.f^ REAR-ADMIRAL SIR FRANCIS BEAUFORT, K C.B. HYDROGRAPHER TO THE ADMIRALTY SIR, The eminent station which you occupy in tlie naval scientific world renders it highly gratifying to me to dedicate the following Work to you as a testimony of my regard and esteem ; while the general accordance of my views on the subject with those of your more experienced judgment, gives me the greater confidence in laying my labours before the Public. 1 have the honour to be, SlT; Your obedient Servant, HENRY RAPER. 225301 PREFACE FIRST EDITION. Tdis Wcrk is intended for the use of all persons concerned either with the navigation of ships or with the determination of latitude and longitude on shore. The present volume, which is devoted exclusively to tlie prac- tice, contains all the rules and tables necessary in navigation, and for the determination of latitude and longitude by means of the sextant or reflecting circle. The study of its contents demands no |>revious attainments beyond the knowledge of the elements of arithmetic. Every endeavour has been made to render the whole easy of reference, and to adapt it to the use of those who may desire to instruct themselves. Rules which admit of more cases than one, as, for example, that for applying the equation of equal altitudes, are given in the form o( tables ; so that the several con- ditions involved, and their mutual connexion, being exhibited to the eye, the computer is relieved from the sense of complication, and the cliance of a mistake is materially diminished. An ample alpha- betical index is annexed, by whicli tlie reader is at once referred to all the inforiuation which the volimie can afford him. Those who have been brought u]) to the sea, and who have experienced the distaste for long calculations which that kind of life inspires, will not hesitate to admit that the only means of inducing seamen generally to profit by the numerous occasions •A-hich offer themselves for finding the place of the ship is extreme VI PREFACE TO THE FIRST EDITION. breviiy of solution. It is not, however, merely as a concession to indolence, that rules should be made as easy and simple as possible; the nature of a sea life demands that every exertion should be made to ? bridge computation, which has often to be conducted in circum- stances of danger, anxiety, or fatigue, and so to separate the several points, that the seaman may be referred directly to what concerns his case, to the exclusion of all other matter. These considerations have lieen carefully kept in view in the rules, in the examples, and in the form and order of the tables. Two kinds of solutions are employed, and, in general, two only; naicely, an approximate method, and a complete, or, as it is called, rigorous, method. The former may often serve in cases of haste, or when precision is not necessary, and will also afford a conve- nient check against the effects of a mistake in the more elaborate metiiod. All the computations are effected by the well-known methods of inspection and logarithms; and as the former, it is presumed, leave but little to be desired in point of expedition, Gunter's scale, or other mechanical methods, are not employed. Sailing on a Great Circle is, in this work, reduced, like Plane Sailing, to Inspection, by means of the Spherical Travebsk Taulr. Convenient rules are given for finding the distance of the land by its change of bearing, and by its altitude observed above the sea-horizon. The seaman will find every necessary information on the subject of local magnetic deviation. The highly useful problem of determining the latitude at sea, by the reduction of an altitude to the meridian, will be found greatly abridged ; and a table is added for the purpose of shewing the limits within which the result may be depended upon when the time at shi|) is in error. This table will bo found, it is presumed, of considerable utility, as it is perhaps from the want of some specific information as to the degree of confidence which it is safe to place in the result, no less than of a short and easy rule, that this excellent observation is almost entirely neglected ; and, in conse- tjiienee, the latitude, when the meridian altitude is not exactly obtained, is too often lost for the day. The approximate solution of the double altitude, as a questioL of Time, will be found, it is hoped, well adapted to general use: since unless tLe latitude by account is very much in error this PREFACE TO THE FIRST EDITION. VU aietnud determines both the true latitiiJe anil tlie time at ship; and the computation of the time is one with which seamen are familiar in the next degree to tnat of the latitude by meridian altitude. The principle is not new, but rules have not hitherto been given for computing directly the error of the latitude by account. The first approximate method of clearing the lunar distance ia new, being effected, like many other problems, by the Siiherical Traverse Table. The rigorous method is a modification of Borda's, and employs five logarithms, of which two only are taken out to seconds. In a work in which many of the methods are new, I have felt it would be more satisfactory to the professional reader to find them illustrated by observations actually taken at sea. The examples are accordingly selected from the journals of Captain W. F. W. Owen, who kindly lent them to me for the purpose; though, neces- sarily, in proceeding by fixed rules, I could not introduce the solutions employed by that distinguished navigator. The remaining observations have been furnished to me by the Rev. G. Fisher, astronomer to Sir Edward Parry's expedition to the Polar Seas. In order to enable the computer to judge of the degree of pre- cision to which he attains, the degree of dependance to be placed on tiie result, or the limit of probable error, is indicated. This is the more important, as very indistinct and erroneous notions prevail among practical persons on the subject of accuracy of com])utation ; and much time is, in consequence, often lost in computing to a degree of precision wholly inconsistent with that of the elements themselves. The mere habit of working invariably to a useless precision, while it can never advance the computer's knowledge of the subject, has the unfavourable tendency of deceiving those who are not aware of the true nature of such questions into the per- suasion that a result is always as correct as the cop puter chooses to make it; and tlius leads them to place the same confidence in all observations, provided only they arc ivorhtd to the same degree of accuracy. By habitually following the short precepts laid down on this point, tlie computer will learn insensibly to estimate the value of liis results ; of which, since the limit of error is the sole ciiterion of the accuracy of any determination, he cannot otherwise be a judge. The degree of precision to which it is proper to carry the work in any case is observed, in general, in the examples. In the Tables every endeavour has been made to repde"- the Vm PREFACE TO THE FIRST EDITION. collection complete for the purposes required, and to compress the whole into small compass. For the sake of clearness, a different figure has beei) adopted for the argument and for the numbers in the body of each table. In the logarithms six places of figures only are employed, because a single result in which six places are necessary cannot be depended upon to the degree of precision obtained. On the same principle, some of the logaritlims are given to three places only. The log. sine square of half the arc, Table 61, universally familiar to seamen in finding the time, is given, for the convenience of this constant computation, to every second of the 12 hours. By means of this term tables of versed sines are dispensed with, all our solutions being either numeral or purely logarithmic. I have not, either in the Rules or the Tables, aimed to make that additive which is in the nature of things subtractive. The precept subtract is as easy as the precept add; and when the student has the natural process before him he may be led to dis- cover the reason of it; and must thus, by attention, always advance in knowledge of the subject. But an artificial process obstructs the exercise of the faculties, or leads the student, who reflects on what he does, to false conclusions. The composition of the Table of Maritime Positions has been a very laborious task, and has caused great delay in the appearance of the Work. The numerous chronometric measures furnished of late years liave rendered it necessary to deduce longitudes in a more systematic and accurate manner than that hitherto followed, which has cliiefly consisted in modifying former determinations by means of those succeeding them. Absolute, or astronomical positions, and relative positions, being distinct things, and the latter being by far of the greater consequence to navigation, it is necessary, preparatory to a complete and final arrangement, to separate these two kinds of determinations. Accoi'dingly, in a series of papers, some of which have been already published in the Nautical Magazine,* I have endeavoured to arrange the chronometric differences of longitude with reference to certain fixed points, convenient for the purpose, which it is proposed to call Secondary Meridians. These standard • The data or evidence for the several positions being given in these papers, the ralue of each determination is easily appreciated ; and accordingly, individuals in pos- session of one or more good watches may, by correcting defective measures, or by establisliing new linlcs of connexion, render material service to maritime geography. See Nautical Maijazine, 1839, and followin;^ years. rUEFACE TO THE FIRST EDITION. IX Iiositioiis, of which the number assumed is eighteen, being con- siderably distant from each other, are determined nearly enough for present purposes, and would, according to the system proposed, be finally settled by long series of astronomical observations. An account of the principles adopted in this arrangement, and of the several voyages and surveys from which the materials have been taken, will be found, together with some suggestions for the advancement of the subject, in the Nautical Magazine. But it is necessary to state here, that the late determinations of the longitude of Madras have, from the importance of that position, occasioned a long and intricate discussion. Mr. Riddle and Mr. Maclear have compared observations of moon culminating stars made at Madias, with like observations made in Great Britain and at the Cape of Good Hope respectively. According to their computations, which agree very nearly, the received longitude, 80" 17' 21", is about 3' "21" too great. The number and superior character of these observa- tions, and the agreement of the results, liave led me to adopt, with- out hesitation, 80° 14' 0" ; while the magnitude of the correction has rendered it indispensable to trace its effects on the longitudes of the Eastern Seas.* Precision in the Maritime Positions, especially in the longitudes, becomes, as navigation advances to perfection, a matter of increasing importance; because, where longitudes are well determined, the error of a chronometer may be ascertained on every occasion of making the land. It will not be out of place to remark here that it is high time the chronometer should be found, like the compass, among the stores of every vessel beyond a mere coaster. It would be superfluous to attempt to prove that the hardships and privations consequent on missing a port, the losses of ships from being out in their reckonings, and the evils incident to navigation generally from the want of a ready means of checking the enormous errors to which the dead reckoning is liable, would, in numy cases, have been prevented by a chronometer. In urging tiiis recommendation, it is, of course, taken for granted tliat they to whose hands the chronometer is entrusted are qualified to make a ])roper use of it. Employed merely as a check, a single chronometer cannot fail to prove of great service ; but too firm a reliance on such an instrument would lead to the dangerous error • The accepted LnngituJe of Madras, India TrigonoiDetiical Survey, 1878 (sea pnge ?94), is 80° 14' S« ' E K PREFACE TO THE FIRST EDHION. i»f relaxing that vigilance which tlie known uncertaintj of the dead re-jkonuig keeps perpetuallv alive. A list of times of high water, or, as they are now called, Esta- blishments of Ports, is not given. The researches on tlie tides made of late years by Mr. Lubbock and the Rev. W. Wheweil, iiave proved that the establishment cannot be truly deduced but from numerous observations, and consequently that a sim])le recorded time of high water is altogether insufficient. Moreover, if the esta- blishment were correctly known, the time of high water, as also the height of the tide, cannot be determined without other elements, which, except in comparatively few places, are not afforded. But in navigation it is not the true instant of high or low water that is required so much as the time at which the flood or ebb stream turns, oecause tliis last affects every vessel when near the shore ; and the proper place for information of this kind is, obviously, the Sailing Directions. Although some results of the kind might be advantageously placed in a general work on navigation, yet the uncertainty of almost all that has been published, and the difficulty of collecting better materials, will, it is hoped, excuse the omission, at least for the present. It may, however, be remarked, that under whatever form it maj hereafter be found advisable to publish particulars of the tides, the observations required are so numerous, the discussions so tedious, and the whole subject so complicated, that no individual could undertake successfully to treat this branch of navigation unless ia a work devoted exclusively to its consideration. The subject of Maritime Surveying, usually treated in works of this kind, has been omitted. Surveying is no part of the navigation of a ship, and a survey having any pretensions to authority can scarcely be made by a person whose qualifications for the task are confined to the slender information contained in a few pages. A survey is a matter of too great consequence to the security of navi- gation to be received from incompetent hands ; and the seaman who desires to acquire a knowledge of surveying should study works treating expressly of this branch of science. The customary chapter on the Winds has likewise been omitted. The subject, generally, does not belong to tfie navigation of a ship,* and, even if it did, the general information contained in a few pages, thougli interesting as a branch of natural philosophy, is PH.KTACE TO THE FIRST EDITION. xl neoessarilv too vague to be effective in shaping tlie course. The same applies to Currents, and also to the Marine Baronietc r ; which, though matters of important consideration in sea-voyages, are not concerned in the practice of navigation, since this term, in btrictness, comprehends only tlie consideration of the place of tlic ship when her circumstances and destination are given. Tlie space gained by the omission of these collateral subjects, and other matters sometimes introduced, is appropriated to the numerous practical details of the proper subjects of such a treatise. The Work will be completed by another volume, wliicli will be entitled the Theory of Navigation, and will contain the construc- tion of the rules and tables, for the advantage of those who desiie to confirm their practical knowledge by matliematical investigation. It will contain, likewise, those methods in which the transit and azimuth instruments are employed. The present volume being thus, in tlie ordinary practice of navigation, indepentlent of the second, no notice of anotlier volume appears in the title-page. By the term Theory is commonly understood, in this particular :nibject at least, the scientific principles on which the rules are formed. Considerations of this kind are thus altogether excluded from the present volume; but, on tiie other hand, that rationale, or process of reasoning, which, in considering the nature of the case, is obvious to common sense or apprehension, is, in raost cases, intro- duced, as necessary to a clear understanding of important points. The theory and the practice are thus kept purposely distinct. The former is not always necessary to successful practice; and rules constructed for ready and general application approacli to perfec- tion in proportion as they leave less to individual judgment or skill. It is the custom, generally, to teach the theory first ; the impression forced upon me is, on the contrary, that the practice is itself the best foundation for sound and rapid advancement in the theory. For he who has acquired the practice knows the nature and extent of the subject; and in proceeding to the theory he has a distinct pev^cption of the object to be attained. This is not the place for a discussion on these points; but it was incumbent on me to state, in a few words, the grounds of the arrangement adopted. It is manifestly the duty of a writer, who undertakes to treat a subject in a thoroughly practical manner, not only to discuss every point which ])resents itself, but also to pronounce a decided (>)iiiiii)n in every case. It is proper to bring this point under tlic notiru nl Xli PREFACE TO THE 1-IRST EDITION the reader, who, especially if he has more experience in these inattprs than myself, might otherwise be disposed to consider many things In this volume as laid down too positively. I cannot close the preface to a work which has been some years in preparation, and in which I have endeavoured to reduce to a practical form every useful consideration which has been suggested by my own experience or by intercourse with eminent officers and men of science, without soliciting the indulgence of the reader to errors and to deficiencies. Absolute correctness, especially in tables, is scarcely attainable ; and in a treatise which contains much that has not appeared before, I cannot reasonably fiatter myself that, notwithstanding every care and attention, some small inaccuracies may not be found. H. B Saptemhkr 1 6'}0 ADVERTISEMENT THIRD EDITION. In tlie Advertisement to tlie Second Edition Iliad the satisfaction of being able to state that the Eoyal Geographical Society had conferred the flattering distinction of their gold medal on the first edition, and that the Lords Commissioners of the Admiralty had honoured my work by ordering it to be supplied to Her Majesty's Navy as ship's stores. The present edition has been greatly augmented. Much of the -work has been rewritten. Two approximate methods of determining the time, though of inferior value, are introduced, since a work aiming to be complete for practice should contain provision for extreme cases. Nos. 789, 791. The introductory portion, it had often been suggested, was in- sufficient for the purposes of elementary instruction. It is easier to allege this, than to lay down a condition which is to determine the extent of such preliminary matter. An attempt, however, has been made to fix a limit, on the following grounds : — The most general defect, perhaps, in the education of seamen, as regards the present subject, is an insufficient knowledge of iiiitlimetic; by which I mean, not of the more advanced rules, but of the elements, and especially of proportion. Now all ques- tions to which arithmetical processes are applied involve some proportion, which the operation is to bring out, or distinctly assign ; and it appears, accordingly, a great omission in our education that we are not more exercised on this point, which is tbe sole object or end of the processes which we learn to practise mechanically! KIV ADVKRTl&EMKiNT TO THR THIRD KUITION. Again, in geometry, it is not tlie variety of problems wliicb benefits the practical man, but a v^ell-grouiided and familiar know- ledge of a few comprehciibive propositions, which he applies readily, and with confidence; and the geometrical knowledge which appears to me to suffice to our present purpose is comprised in, — 1, the property of the square of the hypothenuse; 2, the measure of an nngle at the cir^'iuiference ; 3, the similarity of pl-ine triangles The first is of general importance; the second includea the problem of fixing a station by means of two angles subtended by three objects; and the third is the basis of trigonometry. In this edition, therefore, proportions and fractions are treated at some length, and illustrated by numerous examples which afford tlie student abundant exercise ; and a short course of geometry is given, after the manner of Euclid, sufficient to establish the above important theorems. These limitations, the reader will bear in mind, are intended to apply only to that particular quantity of elementary matter which is assumed to be necessary and sufficient for the scale of attain- ment contemplated in the present volume. In the Table of Positions many points of information of con- sequence to seamen are expressed by means of a new system of Symbols. In these days little apology is required for introducing a scheme which a few years ago would have been deemed a rash innovation. But a growing tendency to the use of symbols mani- fests itself on all sides. Efforts have been made to represent, as far as possible, all matters of instruction under a form addressed to the eye ;* and symbols effect this object in an eminent degree, for their distinct and cons])icuous forms, contrasting with the mono- tonous aspect of alphabetic writing, arrest and fix the attention, while their extreme conciseness admits the insertion of matters to which, for want of room, no allusion could otherwise be made. The employment of symbols, therefore, on a more extensive scale than we have yet been used to, and that at no distant period, may be considered inevitable ; and the present system, which bos occupied my attention for several years, is proposed as so far deserving consideration that it is constructed with rigid adherence to principles.f The number of sigus which I have ventured to • The Physical Atlas Is an example. f The necessity for a uniformity in hydrographic sym'iols has already shewn its.Mf. Symbols similar in character denote, on the French charts, rocks 'ibuce the water, <>ui1 oil the Kussian charts rocks below the water. ATJVF.RTISl'.MKNT TO TlIK THIRD EDITION, \\ iiifrodiipe is small, since, in matters waiting tlie sanction of experi- (Mice. it is better to move too slow than too fast. The introduction of symbols has necessarily modified the original design of the work, as described in the preface, and has justified allu- sion to many matters which otherwise would not have found a place •n it. The chief labour of this eaition tas, indeea, ofthe two former) has been tlie Table of Positions, which, in consequence of the nu- nierous references made to my labours in this country and abroad, I was desirou3 to extend. The list now contains 8,800 places; and as the degree of accuracy is indicated wherever I have found the means of forming a judgment, and as many physical details are supplied, — such as the dimensions of islands, heights, and the depths of shoals, — the table may be considered as representing the state of maritime geography at this day. The number of voyages, charts, and surveys, which it has been necessary to consult, — the labour of digesting and comparing the mass of materials collected, and the introduction, by a new method, of numerous details im- portant to navigation, — will, it is hoped, excuse tlie long delay in the appearance of this edition, and account for the work having remained out of print for nearly three years. In conclusion, I gladly express my obligation to the draftsmen and other gentlemen of the Hydrographic Office, whose patience during many years I have sorely taxed in the inspection and re- examination of thousands of documents, and without whose active and disinterested assistance I must have left much in a very unsa- tisfactory state. ADVERTISEMENT TO THE NINETEENTH EDITION The revision and enlargement of this edition of the « Practice ol Navigation and Nautical Astronomy » was undertaken with con- siderable diffidence, it being felt, that while it was possible to spoil little could be done to improve, this best of practical works ^^ Criptrir Sr golden age of practical Navigation and v.,.+iP.l Survevin- by an officer in constant communication with Ltor aTd HorsbLgh, and the Captains and Masters who served under these distinguished chiefs in England and India LTutliant Eaper's labours are founded u^^n a t^^-o^f P-"^ experience, and may be looked upon as the work of a Sailor for the use and benefit of Sailors at Sea. , One chapter alone required to be re-written. The use of iron in modern Lpbuildhig, by its natural effect on the Mariners Compass, having greatly increased the difficul les of navi^^a.^n at sea some additions have therefore been made to what Eaper Lad Lady written upon this important subject. This chaptei^ as well as^U parts of the book referring to the variation and LTation of the'compass, has been re-written by Captain W. Mayes, R N late Superintendent of Compasses at the Admiralty. _ ■ Captain Mayes has also assisted in making a careful examina- tion of the whole work, which is sufficient guarantee for its having nrs^utfytwedhowwell and earnestly Eaper had c^^^^^^^^ out the intention expressed in the Preface to his First Edition (sTe p. V) of" inducing seamen to profit by the numerous occasions ' Sailors are earnestly requested to read til? Trefoct ADVEKTISKMKNT TO TIIIC NINETKICNTII KUITION. XVU (vbicb offer themselves for finding the place of the ship ; " by laying before them methods whose " extreme brevity of solution abridged computation." These short rules aid the prompt decision upon which the safety and success of a ship at sea so often depend. A brief study of the comprehensive index will call attention to " the numerous occasions " alluded to. The key to most of the modern short methods for fixing the position of the ship will be found in Raper's " Practice of Navi- gation." Under the head of " Degree of Dependance " is placed before the navigator the amount of possible error, a thought which should never be absent from bis mind in considering the estimated position of a ship, with the view of determining his future proceedings. The sailor's attention is earnestly called to the chapter entitled " Navigating the Ship," which contains what John Davis, the navigator, writing in 1607, aptly termed the " Seaman's Secrets." In this, the concluding chapter of the work, Raper shows clearly the never ceasing watchfulness that is required, in both fair and foul weather, in obtaining the observations, terrestrial as well as celestial, necessary to conduct a vessel in safety from one port to another. The simplicity of its mathematical theory makes Navigation appear an easy matter to men teaching or using it on shore ; but Pilotage, common and proper, is a very different business when practised by sailors in a gale of wind, at night, or in hazy weather, on board a ship at sea. Proficiency in the science can never compensate for a lack of experience in the handicraft of naviga- tion. This experience can be attained only by incessant practice at sea; by a capacity for taking trouble, unceasing caution, and a desire to do well. In such labours the sailor will find no better friend and assistant than Raper's " Practice of Navigation." No changes in the numbering of the paragraphs have been made, and great care has been taken to leave the book in the style in which it was originally written, so that old students will have no difficulty in finding the various methods with which they are familiar. Some slight changes have been made in the Tables. Con- siderinrr tlie great increase of speed attained by modern steam- ships, Table 1, formerly Table 2, has been enlarged from 300 to GOO miles of distance. The Table giving the Diff. lat. and Departure for every quarter point has been withdrawn. Xviii ADVERTISEMENT TO THE NINETEENTH EDITION. Table 10, of Maritime Positions, upon which Eaper bestowed a very large amount of labour, has been revised with great care from the latest Admiralty Charts, so that it may still " be considered as representing the maritime geography of this day " (see p. xv). These; positions mainly deiJend on the Table of Longitudes accepted for Secondary Meridians, amended from telegraphic observations to 1887, published in the Admiralty " Instructions to Hydrographic Surveyors." This Table of Secondary Meridians has been inserted in the Explanation of Table 10. Steam having in a great measure rendered Table 12 obsolete, it has been replaced by a Table of the navigable Mercatorial Dis- tances between the principal ports and points of the world. Tables 11 and 13 (Approximate Variation of the Compass, and Tide-hours, or Establishment of the Ports) have been taken out, as the Admiralty Charts, and Admiralty and Indian Tide Tables, published yearly; with the Chart of Curves of Equal Magnetic Variation (No. 2598), corrected up to date; always give the latest information. These tables have been replaced by others showing first: where docks &c. may be found and coals obtained ; and second, the position and nature of the Time signals, in all parts of the world, for the correction and rating of chronometers. Table 65, of natm-al sines, taiigents, &c., to assist magnetic computations, has been inserted in lieu of that of log. sines, tangents, &c., to quarter points. With these few exceptions the Tables retain the same numbers they held in former editions. In conclusion, thanks are due to Captain John C. Almond, Nautical Inspector of the P. and 0. Company, for his many valuable suggestions. THOMAS A. HULL. ilAMEE, Honor Oak : December 21, 1810. In this reprint of the Nineteenth Editiof., the Sun's declination, the Sidereal Time, and the Equation of Time have been given for the years 1901, 1902, 1903 and 1904, in Tables 60, 61, and 62. Table 60a, correction of Sun's declination in Table 60 to 1928, has been restored. Tables 10, 12, and 13 have been brought up to date. Table 38, Corrections of Altitudes of Sun and Stars, has been extended, and the gross corrections are given for ' Height of the eye ' up to 60 feet. Table 47, Limits of the Reduction to the Meridian at Sea, has been recast. Table 70, Logarithms for computing the Reduction to the Meridian at Sea, has been extended to 35' of declination. Tables 41 and .52 have also been recast. CONTENTS. INTRODUCTION. I. Fractions 1 II. PnoPOETlON 10 III. LOOARITHMS 18 IV. Peactical Geomethy 21 V. Geometet and Plane Tkioonometbv 32 VI. MtrTHons of Solution fiO VII. Spueeical Teigonometey 65a NAVIGATION, p I. PFFiNmoNS 55 II. ISSTBUMENTS OF NAVIGATION 63 I. The Compass 63 „ „ Variation of 70 „ „ Deviation of 73 „ „ Adjustment of 81 Magnetic Maps 82 II. The Log and Glasses 104 Sextant, Peotractoe, and Station-Pointer 143 & 178 III. TnE Sailings 106 I. Plane Sailing, with Teaveese, Ccebent, and Windward Sailings 106 II. Parallel Sailing, with Middle Latitude, and Mercatdr's Sailings 119 III. Great Circle Sailing 129 IV. Taking Departdres 137 I. By a Single Bearing and Distance 137 II. Determination of Distance 137 III. Methods by the Chart 143 V. Charts 145 I. Use ok Meecator's Chaet 116 „ Projection of a Geeat Circle U7 II. Oon.itruction of Mercatok's Chart M!) III. Properties of certain Peojections 149 VI. BOLNDINO l.'>l CONTENTS. rxoF The Ship's Journal 154 I. Keeping the Ship's .Tocbnal .. l-'yi IL The Day's Work l-"^" NAUTICAL ASTRONOMY. Definitions 16] II. Instruments of Nautical Astronomy 1-78 I. The REtxECTiNO Instruments, Sextant I'S II. The Aetificiai. Hohizon 188 III. The Cheonometeh 191 III. Taking Observations 1^3 I. Observing Altitudes 193 II. Observations ttith and without Assistants ... 201 III. Employment of the Hack Watch 202 IV. Finding the Stars 203 IV. Subordinate Computations 205 I. The Greenwich Date 206 n. Reduction of the Elements in the Nautical Almanac 207 III. Conversion of Times 216 IV. Houh-Angles 218 V. Times of certain Phenomena 224 VI. Altitudes 230 VII. Azimuths 240 v. Finding the Latitude 243 I. By the Meridian Altitude 243 II. By the Reduction to the Meridian 249 III. By Double Altitude of the same Body 255 IV. By Double Altitude of different Bodies 270 V. lir the Altitdde of the Pole Star 277 VI. Finding tee Tim r, • I. By a Single Altitdde 278 II. By Diffeeence of Altitude near the Meeidlah 285 III. By Equal Altitudes 287 IV. Rating the Cheo.nometeb 233 PAOM VII. PlNDINQ THK LONGITUDE 2t>7 I. By the Cueonometbb 297 II. B» THB Lunar Obseetation 301 m. By the Moon's Altitudb 319 IV. By a» Occhltation 322 V. By Eclipses of Jupiteb's Satellites 3?,fi VUl. Finding the Vabution op the Compass 326 L By THB Amputudh 326 II. By the AzraoTK 328 III. By Asthonomical Bearings 331 IV. By Tkrhestelil Beabings 333 IX. The Tides 335 I. Phenomena of the Tides 33S II. Edles for Finding thb Timb of High Wateb... 341 III. TiDB Obsebvations 345 NAVIGATING THE SHIP. 1. Shaping the Couhse 347 II. Place of thb Ship 3S1 in. Deisemining the Coerent 364 IV. Making thb Land 365 Ekplakation or the Tables 377 TABLES. N^.VIGATIOK Tnii Sailings :— TABLE PAGE P&na 1. Ti-averse Table to Degrees 432 Explanation 377 3. Departiire and Corresponding Difference ofLongitnde 622 „ 381 4. Difference of Longitude and Correspond- ing Departure 523 „ 381 5. Spherical Traverse Table 524 ,, 382 6. Meridional Parts 533 „ 385 Departures : — 7. For finding the Distance of an Object, by Two Bearings and the Distance run between them 538 „ 386 8. True Depression or Distance of the Sea Horizon 539 „ 386 9. Number of Feet subtending an Angle of 1' at different Distances 539 „ 387 10. Maritime Positions • 540 „ 387 11. Places at which Docks may be found, Coals obtained, &c 634 „ 406 12. Navigable Mercatorial Distances 635 „ 406 13. Time Signals 644 „ 407 Tides : — 14. Epacts 646 „ 407 15. Semimenstrnal Inequality 046 „ 407 16. Approximate Rise and Fall of the Tide 646 ,. 407 NAUTICAL ASTEONOMY. Repuctiok of the Elements in the Nautical Almanac:— 17. For converting Arc into Time 647 „ 408 18. For converting Time into Arc 617 „ 408 19. Correction of the Sun's Declination, at Sea 648 „ 408 • Tallin of Longitudes accepted for Secondary Meridians, 392. CONTENTS. XX HI TARLK riOg Ff.r.ti 20. Correction of the Equation of Time, at Sea G48 Explanation 408 21. For reducing Daily and Twelve-hourly Variations 649 „ 408 21a. Logarithms for reducing Daily Varia- tions G5.5 „ 409 22. For reducing the Moon's Declination ... Git? „ 409 2.3. Acceleration 650 „ 409 24. Retardation 659 „ 409 25. For finding the Equation of Second Dif- ferences 060 „ 409 Times or certain Phenomena: — 26. Apparent Time of the Sun's Rising and Setting 661 „ 410 27. Approximate Apparent Times of the Meridian Passages of the principal Fixed Stars 664 „ 410 27A.Correction of the Times in Table 27 ... 665 „ 410 28. Correction of the Time of the Moon's Meridian Passage 665 „ 410 29. Hour-angle and Altitude of a Body upon the Prime Vertical 666 „ 41J A.UITCDE : — 30. Apparent Dip of the Sea Horizon 671 „ 411 31. Mean Astronomical Refraction 671 „ 412 32. Correction of the Mean Refraction for the Height of the Thermometer 672 „ 412 33. Correction of the Mean Refraction for the Height of the Barometer 673 „ 412 34. The Sun's Parallax in Altitude, and Semidiameter 673 „ 413 35. Dip of the Shore Horizon 673 „ 413 36. Corresponding Thermometers 674 „ 413 37. Corresponding French and English Measures 674 „ 413 38. Corrections of Altitude of the Sun and Stars 675 „ 41i 39. Correction of the Moon's apparent Alti- tude 676 „ 414 40. Corresponding Horizontal Parallax and Semidiameter of the Moon 685 „ 414 b 41. Correction of the Moon's Equatorial Parallax for the Figure of the Earth . 685 42. Augmentation of the Moon's Semi- diameter , 685 43. Correction for reducing the true Alti- tude of the Sun or a Star to the appa- rent Altitude 086 44. Correction for reducing the true Alti- titude of the Moon to the apparent Altitude GSG 45. Parallax in Altitude of a Planet 6SG 46. Azimuth and corresponding Change of Altitude in One Minute of Time 687 LatITUL'K : — 47. Limits of the Rtxluction to the Meridian at Sea 688 48. Value of the Reduction at which the Second Reduction amounts to 1' G88 49. For computing the Reduction to the Meridian in Seconds 689 50. For computing the Second Reduction in Seconds 691 51. Correction of the Altitude of the Pole Star 692 52. Reduction of Latitude 693 liONOnUBE: — 53. Correction of the Lunar Distance for the Contraction of the Vertical Semidiameter 693 54. Error of Observation arising from an Error of the Parallelism of the Line of Sight 693 55. For correcting the Lunar Distance for the Spheroidal Figure of the Earth... 693 5C. For computing the Moon's Second Cor- rection of Distance 694 - 57. Correction of the Greenwich Mean Time for the Second Difference of the Lunar Distance 695 B8. Error of the Ship's Place in Nautical Miles, and of the Longitude in Time, corresponding to an Error of 1' in the Lunar Distance t'>95 Explanation 414 » 415 415 415 415 415 41 C 416 416 417 417 418 418 418 CONTENTS. XXV Variation of tub Compass: — 59. Amplitades 690-7 Explanation 419 59a. Correction of the Amplitude observed on the Horizon, for the Effect of Re- fraction 690-7 „ 420 To Supply the Place of tub Nautical Almanac : — 60. Declination of the Sun 698 „ 420 60a. Correction of the Sun's Declination 700 „ 420 01. Sidereal Time and Riglit Ascension of the Sun 701 „ 421 62. The Equation of Time 703 „ 421 6:{. Mean Places of Stars 705 „ 421 LodAKI THM.'' : — 64. Logarithms of Numbers 706 „ 422 64a. Spheroidal Tables 724 „ 423 65. Natural Sines, Cosines, Tangents, Co- tangents, Secants, and Cosecants 725 „ 4?3 66. Log. Sines of small Arcs to each Second 726 ,, 423 67. Log. Sines of small Arcs to Ten Seconds 735 „ 424 68. Log. Sines,Cosines, Tangents, Cotangents, Secants, and Cosecants 738 „ 424 69. Logarithm of the Square of the Sine of HalftheArc 828 „ 426 70. Logarithms for computing the Reduction to the Meridian, at Sea 803 ,, 427 71. Logarithms for computing the Correction of the Latitude bj Account 897 „ 423 72. Logarithms for computing the Equation of Equal Altitudes 899 „ 428 73. The Logarithmic Difference 900 „ 428 74. Proportional Logarithms 909 „ 129 Aur.RKviATiONS adopted on Admiralty Charts, WITH Explanatory Noif.s 9?li Gknkral Indlx 927 Maps of botu IIemispueuls To be ^'laced between pages S2 mni 83 DlA'lBAM TO ILLUSTRATE SUMNEU's METHOD FOR FINDINO THE Posnioy OK A Ship at Sea Tofacepaye 363 INTRODUCTION. r. KitACTiONS. II. Proportion. III. Logarithms. IV. Prao- TiCAL Geometry. V. Eaising the Trigonometrical Canos. VI. Methods of Solution. 1. Vulgar Fractione. 1. A NUMBER which is a portion of 1, or unity, is properly- called a fraction ; thus, if we divide a foot into 3 equal parts, each of such parts is the fraction called a third, and written ^. These numbers arise, in arithmetical operations, in division, when the dividend is not divisible by the divisor in whole numbers, or, as they are called, integers; thus, if we divide 10 feet into 3 equal parts, each will measure 3 ft. and one-third, or 10 divided by 3 gives the quotient 3, and 1 over — that is, 1 not divided like the rest ; but proceeding now to divide this 1 by 3, we call the result or quotient ^ ; that is, 1 divided uy S. 2. If we divide 1 into four equal parts, each is one-fourth, written ^ ; if into 5 equal parts, each is vue-fftii, written ^; thus, the name of the fraction is that of the number of parts into which the unity or entire quantity is divided ; and this number is hence called the denominator of the fraction. 3. If we take two of thi-ee equal parts of subdivision, or two- thirds, we write § ; if we take three of four equal subdivisions, we write I ; if we take three of seven equal subdivisions, we write ^ ; and 80 on : the number 2, 3, in these examples, which shews or enumerates the number of fractional parts taken, is hence called the numerator. The term fraction is thus used to denote not only one part or subdivision, but any number of such. 4. In enumerating fractional parts we may go on, for example, s' l> 5> I' I' 5' 5' '^''- Here f represents the whole, or entire quantity, since it enumerates as many parts as the whole is divided into; the fractions (so called) beyond this, as 9,, i, are all greater tlian 1, and are termed mixed or improper fractions. 5. The fractions to the left of f are less than 1, and are proper fractions; hence, when the numerator is less than the denominator, the fraction is less than 1 ; when equal, the fraction represents 1 ; and when greater, it is greater than 1, and is capable of being resolved into a whole number witl or without a fraction. INTRODUCTIO>f. Hence aUci, the greater the denominator the smal.er the fractioa, and the smaller the denominator the larger the fraction. 6. If we take a line AB, and divide it into 3 equal parts by tiie points K, l; and another line BC equal to it, and divided similarly Rt M, N, then Al is « of AB, or of i. Then the parts being all equal, Ak and kl, are equal to lB and Bm, and these to mn and nC; therefoie Ak and kl are J of AC, that 18, of 2. Hence Al is | of 1, and ^ of 2 ; or, ^ of 2, and | of 1 are the same thing. If AB is 1 yard, it is evident at once, since 2 ft. or I of 1 yard are | of 6 feet, or 2 yards. 7. The value of a fraction is not changed by multiplying the numerator and denominator by the same number. The term one-half is equivalent to two-quarters, to four-eighths, and so on ; that is ^, f , J, &c. are all equal ; since it is evident that the result is the same if we divide the whole into twice the number of parts, and take twice the number, or into 3 times the number of parts, and take 3 times as many of them. The above fractions are ^, the numerator and denominator being both multiplied successively by 2. Again, take f, multiply both numerator and denominator by 3, it becomes -j*^^ : if now we take a line and divide it into 5 equal parts, and 15 equal parts, it will be the same thing whether we take two OJ the larger parts, or six of the smaller, wliich are J the size. 8. The value of a fraction is not changed by dividing the nume- rator and denominator by the same number. This appears in exactly the same way as the above, in any case ; thus, -/j, dividing both im- merator and denominator by 3, gives |. The process is equivaleiii to dividing the unit into larger portions, and taking fewer of them in proportion Fractions are tlius often simplified : example, -f^j is evidently re- ducible to iVs ; tVjs to Jj.* • A fraction is reduced to its simplest terms by finding tlieit greatest common meayi.-'e, thit is, the largest number which will divide them ba"j without a remainder. To find tif greatest common measure of two numbers, Divide the greater by the less. Consider the remainder as a new divisor to the former divisor as a dividend, and find the next remainder. Consider the last remainder as a ne.v divisor, and find the next remainder, and so on. The last divisor is the number required. If the last divisor is 1, the numbers have no common measure but 1, that is, arc not further reducible. Ex. I. Find the greatest sate of 24 and 124. 24)124(5 J 3 [1] Reduflion to a Cmnmon JJmnmJnatnr. 9. Suppose it is required to add togetlier 5 and |; if wc could p.i once express tliirds in fifths, or fiftlis in thirds, we siiould then merely enumerate tiie number of parts ; but as one of these fractions is no exact number of times g;reatcr than the other, (as may be seen by dividing a line into 5 parts and 3 parts), we cannot do this. But by multiplying the numerator and denominator of one by some number, and of the other by some oilier number, (which leaves the fractions unchanged in value. No. 6) we may select such multipliers as will produce the same number in the denominator; thus, multiplying the numerator and denominator of f by 5, gives ^-y, and muitiplying the numerator and denominator of | by 3 gives ^, and the fracuoi.a f and f are thus reduced to 15ths. Again, to reduce -^^ and ^^ to the same denominator, multiply the numerator and denominator of j'j by 11, which gives ^^^^, and -,', by 12, which gives ^^.*j. These reductions are effected by multiplying each numerator by the other denominator, and the two denominators together ; and the same ajjplies to three or more fractions taken in succession. Hence the Rule: Multiply the numerator of each fraction by every deno- minator, except its own. for the new numerator, and multiply all the dtiiominators together for the new denominator. &.. I. Reduce -, -'-, and -. 3 >5 7 2«.,«7 iM"? .MX'5 „"o »i il 3x15x7 3x15x7 3x15x7 315'3'5'3'S" RI.2, Reduce ii, I. and i. 11x1x7 1x17x7 4x17x2 154 1,9 ,36 17 X a X 7' 17 X 2 X 7' 17 X 2 X 7' 238' 238' 237 SX.S. Reduce-.and-;. iM 175 360 420' 4S0' 4»o' 10. The process of reduction to a common denominator is often necessary in the comparison of two fractions, to find which of the two is the greater ; thus, to compare j'^ and ^, these become ^^| and ^-^ hence -j'f is the greater. 1 1 . Whole numbers are written in the fractional form by em- ploying 1 as the denominator ; thus 3 is written ^, the 1 is in the place of the unit divided into 1 part (No. 2), that is, left entire, and the 3 denotes that 3 such jiarts are taken (No. 3). 12. By means of this last notation whole numbers are reduced to fractions w ith the same denominator, by the rule No. 9. Thus 1 1 and I, or '-' and { hcconie V *"<^ l- 13. Reduce the fractions to a common denominator, add the numes rators (No. 9), and under the sum place the common denominator. Ex. 1. Add tugeUier -'- «d -. These bea,ite ^-^^ = -2, ,.nd ^^^ =■ ^■, ihcs ,m 17 3 17x3 51 3x17 c. r>f.. I. ;-..:. *2 INTEODUCTION. Add together -, -, and -. Ane, Add — , -, and . Ans. • 3 ^ , ' A 364- -^, --, and -. Ans. ^-r^ : 10' 16 3 4S0 [3.] Subtraction. 14. Kule: Reduce the fractions to a common denominator, and subtract the lesser numerator from the greater for a numerator. TiiU!^, suppose it required to subtract J from i, these become Jj, and ■/j, and fg from -3^ leaves Jj, tlie remainder required. Hence it appears that the ditFerence between ]- part and I j)arl )s -\, of the whole. Fhid tl.e difference between ^ and -. These become 1^ a udii 35 35' Subtract -L fro. i-. An. ^. Subtract— from — . Ans. ^ = .^. 13 5 65 6s [4.] Jitultlplicatluii, 15. To multiply a fraction by a whole number is to repeat tiiu fraction a given number of times ; that is, to multiply \ by 3, or to take \ three times, gives \. Hence to multiply a fraction by a whole number is to multiply the numerator. Hence a number multiplied by a (proper) fraction is diminished; thus, 3 multiplied by i, which is ^, is less ttian 3. 16. To multiply a fraction by a fraction, as for example ^ bv |. Since I is the same as twice one-fifth, we have to take J of |, and double the result. To take \ of 4 is to divide } into 5 parts and take one of them; now } is 3 X J^ (by No. 6), and dividing 4 into 5 equal parts gives j,'^-, since 5 such parts repeated 7 times make up 1. Hence 3 of these parts (or 4 divided into 5 parts) is /y, which is therefore -^ of 4- and -Jj doubled, or ^3^, is j of 4. Now, the numerator 6 is the product of the two given numerators, 2 and 3 (as apj)ears by the process) ; and the denominator 35 is the product of the denominators 7 and 5. If we had to multiply this re- sult by a third fraction, the process would be the same ; hence the Kule. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. El. 1. Multiply i, -, and -. Ans. -ii. Ex.2. Multiply |-", bv -. Am. -^ 3 5 7 105 ' • 63 ■ 7 44' Ex. 3. Multiply ij- -. i'nd -. Ans.—. ' ' j6 3 5 S40 17. If we multiply j by itself, we liave *, tbis again by \ gives /^i cow /^ diH'crs little iVoni -^^, and /j is otiual to *, \\'\nci\ is very FRACTIONS. 5 Qiiitli loss tliaii 5. Again, \ multiplied by itself is ,'j, ami tiiis niultiplicd again by 4 's ^^. Hence a proper fraction is diminislied by continually multiplying It by itself. [b.] Dinsion. 18. To divide a fraction, as J, by a whole number, as 4, is to find a new fraction which, repeated 4 times, shall produce ^ : that is, «e have to divide a third into 4 equal parts. It will be at once seen, on dividing a line into 3 equal parts, that to divide each third into 4 equal j)arts, is to divide tiie whole line mto 12 equal parts, and since 4 of such parts, or twelfths, constitute a third, j'j is the required fraction. Hence, as similar reasoning applies to any other traction or whole number, the most general rule for dividing a fraction by a whole number is to multiply the denomi- nator by the given whole number; but if the numerator be a multiple of the divisor, it is better to divide the numerator as it leaves the result in a more reduced state. 19. To divide a whole number, as 3, by a fraction, as -J. Dividing 3 by 1, that is, finding how often 1 is contained in 3, gives 3. Now, it is easily seen, since -J is 4 times smaller than 1, that it must be contained in 3, four times oftener, that is 12 times; and 12 is the product of 3 by the denominator 4. To divide 3 by |. Since | is twice ^, we have to divide 3 by j, and take half the quotient; and we know that to divide by the product of two numbers, 2 x ^, is the same thing as to divide by theiu separately, that is, 3 divided by f is 3 multiplied by 6 (No. 181, and divided by 2 ; or 3-i-| is the same as 3 x f , or ^. Here f is the fraction | inverted. As similar reasoning applies to any numbers and fractions. Me have the Rule. To divide by a fraction, invert the fractional divisor, and proceed as in multiplication. 20. To divide a fraction by a fraction. We have evidently io treat the dividend as a whole "number, and apply to the divisor the rule above. Ex 1, DinJc-^by-. J-^l = - = l. Ex.2. Divide Uy - Aris-ii. Ex. 3. Divide * by -5-. Ans. ". Hence it appears tliat the smaller the fractional divisor tb^ greater is the quotient. 21. When a quantity is both multiplied and divided by the saiue number, it remains unchanged. Hence when the same number occurs in the numerator and denominator of a fraction, or of two or more fractions multiplied together, we simply omit or erase it ; a.s, m^, '.1-' '♦k'kZ -1 ' ' ' ± (, I IKTBOD'JCnOK. II. Decimal Fractions. '12. Tenths, hundredths (which are tenths of tenths), and so oii, are called Decimal Fractions, and may be written as fractions, havina: for denominators 10, 100, &c., thus, one-tenth, J^ ; three hundredtlis, j-g-^, &c. But as these quantities are counted by tens, like common numbers, it is simpler and more concise to write them in continuation with the common numbers, only taking care to put a dot, called the decimal point, where the whole number ends and the fraction begins; that is, between the unit and the tenth : thus, '21*3"i signifies 21 and 3-tenths and 2-hundredths ; 432-9 signifies 432 and 9-tentlis ; 33-05 signifies 33, no tenths, 5 hundredths. 23. In the fractional part beyond the dot, each figure may be read in its separate denomination, or the whole may be read in the deno- mination of the last: thus, -32 is read either as 3-tenths and 2 hun- dredths, or as 32-hundredths; just as 32 is road either as 3 tens and 2 units, or as 32 units. 24. As-5, (or5-tentlis)isthehalfof l,so-05 is the half of 0-1, or 5 hundredth-parts are the half of one-tenth ; 5 thousandth-parts are the half of a hundredth-part. The half of 5 tenths is 2 tenths and half a tenth, that is, 2 tenths and 5 hundredths, or 025. Hence the fractions, quarter, half, and three-quarters are written in decimals, 0-25, 0-5, and 0-75. All the preceding rules apply equally to decimal fractions , but R8 these last, from their denominators being multiplied by 10, are o/ a uniform kind, special rules have been made for them, relating, however, almost entirely to the placing of the decimal point. [1.] Addition and Subtraction, 25. Place the quantities so that their decimal points shall be it the same vertical line ; for then the quantities of the same denominn- tion will stand together. Then proceed as in the addition or subtraction of whole numbers. n 3rS subtract ii-6i. Add together 0-35, 47-4, and Ex. 0-35 47-4 Sura 56-87 Add together 71-99, 4-1, aud Ex. 72-99 4"i . 5 -'3' Snm 129-40 [2] Mult iplication. From 423-5 subtract 97-9. 4-3-5 97-9 26. Multiiily the numbers together as whole r.uLnbers, and poii.i ofJ'aa many decimal places in the ])roduct (beginning at the right) as there are decimal places in the multiplier and nmltiplicand together. WIiP!! tlie Jeciuiiil places to be pointed off are more 'ii nuinhei tliiin the figures of tfie product, make up the proper iimiil)cr by prefixing ciphers to the product. Ei. 1. Multiply 34- 1 by 3-71- 34-11 3"72 68z2 »3*77 Multiply '20 1 by "06 The profluct of 201 by 6 is 1206 ; in soi are three decimals, in 06 are two ; to inak.-! up five decimals, a cipher is prefixed £0 1206. In 34-11 are two decimals; in 3-72 are ^co ; therefore four decimal places are pointed 3lT. Ex. 3. Multiply 9001 by 0.034. Ans. 3-06034. ■Rx. 4. Multiply together 1-3, 1-2, and 0-09. Ans. 0-1404. [3.] Division. 27. Divide as in whole niinibers. The rule for placing the deci- mal point is, that the quotient and divisor together must contain as many decimals as the dividend.* Ex. 3. Divide 2-392 by 4-6. 4-6)2-392(52 230 92 9» Here 2-392 contains three decimak, .ind 4-6 one, the remaining tw-o required must therefore be obtained by pointing of! botli figures of 52 thus, -52. Ex. i. Divide 338-4 by 9-4. 9-4)35S-4(36 282 564 Here the dividend has one decimaJ, and the divisor also one, or as many, and the quotient is therefore an integer. 28. When llie dividend has no decimals, ciphers must he annexed, preceded by the decimal point. Ex. 1. Divide 19 by -04. Ex. 2. Divide 132 by 0-7. Annexing two ciphers to 19, gives the Annexing five ciphers (decimals) gi\es COmplele quotient 475. quotient 1885714. Then the n-amher wnirh Ided to one decimal in 0-7 to make up fiv3, four. Ans. 188-5714. 29. AVhen the number of figures in the quotient is not siifHcient to make up the required nuinher of decimals, cijihers must be pre- fi.xed. Divide 17-34 by 3-4. 3'4)"7'34(5' e 17-34 contains two decimals, 3-4 1 only one ; therefore 5 1 must contain aining one required, and be written i?:t. 2. Divide 541-2 by 66. 66)541-2(82 528 132 132 Here 54 J -2 contains one decimal, 66 none ; ice ^2mii8tcontain one, and be written 8-2. • It is always caay to verify the <iuoticnt, since multiplying it by the dvisor should re- produce the ilividend : thus, in Ex. 1, 5 1x3-4 gives (by No. 26) 17-34. The learnei should also exercise his common sense on the results as a security against gross mi.>itakes \ thuii, 17-34 divided by 3 1 will be near 17 diviilcd by 3; that is, less than G (as 5 1 i^^ ^gain, 2-392 dividsd bv 4 6, i« not frj fron 7 diviaed bv 4, (,r » hilf (which is marly -52) El. 1. Divide •1734 by 3'4.. 1 Ki. 2. Diride 2-391 oy 46. Here •1734 lontains four decimal!, and [ Here 2*39i contains three decimals, and |. one; the quotient 51 (Ex. i, above) j 46 none; the quotient (52) must contain ; only two figures, and three are re- three, and becomes o'052. juired , hence 51 must be written 0-051. | Ex.3. Divide 27-9 by 002. Annexing one cipher, the quotient is 1395. Ex. 4. Divide 0-0296 by 5-2. Annexing two ciphers gives quotient 569, which is 0-00569, lince the five in this added to one in 5-2 make up six. 30. Tlie division may always be carried to any degree of accuracy by annexing cipliers to the dividend, as is seen in E.\. 2, No. '28. 31. The decimal point maybe removed altogether from both the divisor and dividend, by continually multiplying each by 10; for the ([uotient will thus remain unaltered. No. 7. The first decimal in the quotient will then appear only with the first cipher annexed to carry on the division. Ex. Divide 27-9 by 0-02. Multiplied by 10 they become 279 and 0-2 ; multiplied again Uity become 2790 and 2, the quotient of which is 1395. This easy process furnishes a complete security against wrongly placing the decimal point in the quotient. [4.] Reduction. 32. The great convenience of aeciraals makes it often desirable :o reduce vulgar fractions to the decimal form. To reduce a Vulgar Fraction to a Decimal Fraction. Divide the numerator by the denominator, adding ciphers bjs required. The quotient is the decimal required. Ex. 1. Reduce - to a decimal fraction. Dividing 10 by 5 (the cipher being added") we find -is 0-2. 5 Ex. 2 Reduce - to a decimal fraction. Dividing 10 by 3 gives 3; the next cipher added gives another 3, and so on continuilly. The fraction required is therefore 0-333, *c. Ek. 3. Find what decimal of i (nautical) mile is 700 feet. niere are 6080 feet, nearly, in i such mile ; hence i foot is of i mile, and 700 f-Vit 700 ftm T-T" of 1 miie, which gives 0-115 of i ""il^t nearly Ex. 4. Find what decimal of i minute is 42 seconds. I second is — of i minute, hence 4: seconds are — or 0-7 of a minute ; or, as it may be Ex.5. Find what decii nalof I foot i 8* inches First, J fcence 875 is 0-75 inches of I inch are *'", , hence Scinches are 8-7 or 0-729, of J foot. Ex.6. Find what decimal of I degre is 8' 37". 37" are r>f ',oro-6 of 1' them '^ro"f'^ Kx. 7. Find what decimal of i day i ^.2"' are ^ o'' i*", or o''.y ; and i' I*- 154106, Sic. 33, Or, reduce tlie given quantity to tiie lowest of its denoiuinatioii-) when tliere ;iro more than one, Jiiui also the integer to which it is referred, to the same dcnoininatii)n ; then divide the given quantity by the integer thus reduced. Ex. 1. (Ex. 3, above.) The given qu.intity, 700 feet, being all in one dfnorainatlon, requires no further reduction. The integer I mile, reduced to the same denomination, is 6080 feet ; then 700 divided by 5o8o gives 0-115. El. 2 (Ex. 5, above.) S inches and 3 quarters are 35 quarters ; and i foot reduced to the same denomination, is 48 quarters ; tlien 35 divided by 48 gives 0-729 34. To reduce a Decimal Fraction to a Vulgar Fraction. Note the number of parts which the unit or integer of the given quantity contains of the ne.xt inferior denomination, and multij)ly the given decimal by this number ; the product is the given quantity expressed in that denomination If this product have a decimal part, multiply this decimal by the number of parts which the unit of the present denomination contains of the next inferior denomination to that just before employed: this product is the quantity which the given decimal contains of that next denomination. Proceed (if there till be decimals), m like manner, to the lowest denomination in which the decimal is required to be expressed Kk. 1. Find the number of feet in o-i i c of i mile. The next inferior den is here feet, of \ Ans. (in the lowest denomination required) 6994 feet. Ex. 2. Find the number of seconds in 0-7 of i minute. The next inferior denomination to that of minutes | ° J is seconds, of which the number in i minute is) — Ans. 42-0 Beconda, the number of inches and eighths in 0-48 of i foot. 0-48 Tlie next inferior denomination to that of feet ) is inches, of which the number in i foot is) '--^ The next proposed inferior denomination to inches ) ' ^„ ''"'''^'* is eighths, of which the number in i inch is) Ans. 5 inches and 6-08 eighths, or - nearly. Find the number of minutes and seconds in o'-- 6-(j8 eighti^ Tlie next inferior denomination to that of degrees! °'^'J is minutes, of which the number in 1" is) ••■•• E^ The next inferior denomination to minutes 1 *+ '°+° nds, of which the Ans. 44' 2"-4. J. ^00 Find the number of hours and minutes in 0-37 of a day. The next inferior denomination to days is) °'" hours, of which the number in i d.iB24j i* 71ie next inferior denominntion to houn u{ »-88 hours minatea, of which the number in 1'' ii 60 ) Ant. 8'' si^-g. jiSo 10 INTRODUCTION. 35. When we propose to use tlie nearest whole nmiiber, rejecting tlie decimals, ii'the decimal is less tliaii '5, we omit it, if greater than •5, we count it as a unit. For example, if we propose to take 31'3 as a whole number, we call it 31 ; if we propose to take 31-7 as a whole number, we call it 32. The reason is, obviously, that 31-3 is nearer to 31 than it is to 32, whereas 31'7 is nearer to 32 than it is to 31. In like manner, we may abridge the decimals themselves when accuracy is not required: thus, for ex. 11 '567 may, when two places only are required, be written 11-57, or when one place only, 116* II. Proportion. 36. By the term ratio we commonly understand the relative niangnitude or quantity of two things of the same kind ; thus, when we speak of the ratio of two numbers, 12 and 4, we mean their relative magnitude, or the result of comparing them together in respect of quantity. 37. The most distinct and intelligible notion which we can form of the degree in which one quantity or magnitude is greater than another, is the number of times one contains the other; that is, the quotient of one by the other is the measure of the ratio. Thus, to compare 12 and 4, we find that 12 contains 4 three times, or the quotient i^-, or the number 3, is the measure of the ratio of 12 to 4.+ (1.) The sign + , caWei plus (which is the Latin for more), signifies adiilive, or to be the Latin for less), signifies stilili-aclive, or t<- Ex. + 3 signifies 3 to be added, —3 signifies 3 to be mbtracteii (3.) The sign x signifies multiplied by. Ex. 7x5 signifies 7 multiplied by 5. (4.) The sign -i- signifies divided by. The operation of division is also indicated by writing the divisor under the dividend, with a line between them. Ex. 14 -r 2 signifies 14 divided by 2 ; which is as frequently denoted thus, — . (.'p.) The sign = signifies equal to (or amounting to). Examples of the preceding, with the results in each case, will stand thus : — (1.) 14 and 3 = 17, or 14+ 3 = 17. (2.) 10 - 3 = 7. (3.) 7 "5 = 35- (•<•) .4-^ = 7, or^=,. These processes appear much more conspicuous to tlie eye than wiien written out in words at length. + But, instead of saying that the absolute number 3 is tlie msseure of the ratio 12 : 4, it is more correct to say that the measure is itself the ratio of 3 : 1 ; because, in all cases of measure, we employ a convenient quantity of the same kind as a unit, as 1 foot, or 1 mile. for length, 1 second for time, &c. ; so the measure of ratio is itself a ratio, but of the limpU-st form that can l)e found PROrOUTION. 11 The ratio or proportion (for tlie teniis are often used indifferently) of two numbers, as 12 and 4, is written thus, 12 : 4, or, as above, >/. 38. Suppose it recpiired to find the ratio of 12 to 5. 12 contains 5 more tiian twice, but not three times. By actual division, ^ gives 25 ; but this, instead of being simpler, is more coinph.x than y. Hence, as we cannot snn]ilify this fraction (12 and 5 having no common measure but 1), it remains as the measure, or represents the ratio of 12 : 5 39. In the same manner is represented tlie ratio of 4 to 12, in which the smaller term is taken first; for though 4 does not contain 12, yet it contains the third part of 12, so that there is still an exaci relation between the numbers in this order : in other words, the ratio of 4 to 12 is the same as the ratio of + to 1 ; but the ratio of J to 1, or a third to the whole, is the same as that of 1 to 3, since each contains the other three times. Hence, 4 : 12, or a : 1, is the same as 1 : 3, or ^ the same as ^, which is the measure of j%. 40. There is an employment of ratio or fractions which is often embarrasiing to unpractised arithmeticians. If we increase 6 to 7, we add [sixth, for 1 is -^ of 6, and 6 + 1 make 7 ; but, if we now diminish 7 to (5, we take away \-seventh, for 4 of 7 is 1, and 7— 1 is 6. In the first case, we take a fraction of 6, in the second, a fraction of 7 ; and it is obvious that the same quantity cannot be the same fraction of two different numbers. In like manner 3 increased by -J of itself becomes 4; but to pass back again from 4 to 3, we must take away ^^ of 4. 41. It may be convenient to express the change of a quantity in any ratio, by means of the increase or diminution it undergoes, measured by a fraction of itself. To increase a number in the ratio of f . | is composed of | and *, or 1 and f ; hence the number is to be increased by ■§ of itself. To diminisii a immber in the ratio of }. t is equivalent to f, deducting |, or to 1— y; hence the number is to be diminished by ^ of itself. Ex. 1 . A number is increased in the ratio of — , by wliut fraction of itself is it increased ? Answer I^ " '55' Ex 2. A number is diminished in the ratio of—, by what fraction of itself is it diminished ? Answei, — . 5' 42. The first of two terms taken in order is called the antecedent, Hud the second the consequent: thus, in 12 : 4, 12 is the antecedent, and 4 tlie consequent; in 4 : 12, 4 is the antecedent. 1. Direct Proportion. 43. When two pairs of terms occur, each antecedent having ths Bame ratio to its consequent, the four terms corstitutc an analogy, or proporti()n, as it is also called : thus, 18 and 6, 12 and 4, each pair 12 INTRODUCTION. Iiaviiig for its measure the ratio |. form tliis proportion — 18 is to 6 as 12 is to 4 ; or, as it is written for abbreviation, 18:6:: 12:4. The same is also written thus: 'g" = '^, anil read " the ratio of IS 10 C) is equal to the ratio of 12 to 4."* 44. In every proportion tlie product of the two extreme terms is equal to the product of the two mean (or middle) terms: thus, in 18 : 6 : : 12 : 4, 18x4 = 6x 12 = 72.t This property affords the test by which we learn the various alterations that may be made in a proportion, the original proportionality being still preserved. 45. The following- variations in the order of tiie four terms of a pro)3ortion occur the most frequently : — Given form, 18 ; 6 ;; 12 ; 4 / 4. ; 6 :; 12 C i5 Alternately, 18 : 12 ;; 6 ; 4 , ... 6 ; 4 :; 18 : 12 Ueversing, 6 : ,8 :; 4 ; ,1 I" l.ke manner, „_ . ,^ . . ^ . g • Or, 4 : 12 :; 6 ; 18 ' 12 : 4 ;: i8 : 6 4ti. In a ])roportion, either of the mean terms is equal to the pioduct of the extremes divided by the other mean. Also, either of the extremes is equal to the product of the means divided by tiie other extreme; as in Hence, if any tliree terms of a proportion be given, the fourth may be found. 47. It is often required to increase or diminish a quantity in a certain ratio, or proportion. For e.xample, to increase the number 12 in the ratio of 3 to 1, is to multiply by 3. For the increased (|uantity (which, being yet unknown, we will call .t) is to be to tlie given quantity 12, as"3 to 1, or .t : 12 : : 3 : 1. Whence (No. 44) lxa:=I2x3. Again, to reduce a number, as 13, in the ratio of •5 to 7, is to multi])ly it by 5 and then divide by 7, for the required number (.r) is to be to the given number (13) as 5 is to 7, whence 13x5 ^ = -T-- For example, if certain provisions last 122 men a given time, it is evident that, in order to last 146 men the same time, they must be increased in the ratio ol 146 : 122; tluit is, multiplied by 146, and then divideil by 122. Again, if certain provisions suffice 106 men, and they are required to serve only 74 men, they may be diminished in the ratio of 74 to 106 ; that is, x 74-;- 106. * Hence proportion is also desmbed as being the equality of ratio. t Hence, also, when the products of two pairs of numbers are equal, the four numbers may be written as a proportion. Ex. 22 x 66 = 4 x 363 ; hence 22 : 4 \\ 363 : 60. Csre must be taken in tlie order of the terms, which, though indifferent in a product \s every tiling in a proportioa. 'I PROPORTION. la [1.] Rule Iff Three, Direel. 48. Numerous arithmetical questions occur in a form more or less like this : if 5 men do 20 yards of work, how many yards will 1 1 ineji do, in the same time, and under the same circum- stances.* (1.) The most obvious and natural method of solving such ques lions is the Method of Unity. Tlius, if o men do 20 yards, 1 uian alone will do 4 yards, and therefore 1 1 men will do 1 1 times 4 yards. (2.) The General Method is to arrange the terms in the manner of a proportion, and then to find the unknown term from the other three, (No. 46). Thus, it is obvious that a constant proportion obtaining between the men and their work, we have 5 men ; 20 yards : : 11 men : number of yards required. This process is called the Rule of Three. (3.) They, however, who are practically familiar with ratio, 01 proportion, perceive, on considering the question, the ratio in which one of the given terms is to be changed, so as to suit the conditions ; and thus the solution is effected at a single step. Thus, in the above ([uestion, it is evident that the given number of yards, 20, is to be increased in the ratio of 1 1 : 5 ; that is, in exactly tiie same ratio as the number of men is increased. The solution, therefore, is com- prised in these figures, 20 x V, which gives 44. 49. Various precepts have been suggested for ensuring a correct order in the arrangement of the terms, or the statement of the question, as it is called ; and one of sucii, which is often useful, is t. consider the terms given as standing to each other in tlie relation of cause or agent, and effect (as, for instance, tlie men in the above example and their work). By this supposition (which, however, is arbitrary and unsatisfactory enough in many cases), the four terms are rightly paired, or the antecedents and consequents rightly taken. But the fact is, tliat no mechanical rules can so completely supersede the notion of proportionality as to absolve the mind from all neces- sity for estimating it; and, consequently, the student, if he clearly understands proportion, depends upon it alone; and if he does not, he cannot, from any number of precepts, feel the least confidence in the soundness of his result. As a right apprehension of proportion is most essential to every one who has any thing to do with calculation, we have, for the sake of exercise, solved several examples in each of the above tliree forms. E«. 1. A steam-vessel consumes 13 tons of coal in ij days ; how long will 98 tons last ? (1.) Method of Unity : 13 tons in i}d. or Jd.,is i ton in or ^jd., and 98 ton» ii 4x13 98 X |j or 135'j days, or 13d. fh. nearly. * In the application of the njeg which follow, the circumstances are supposed to remain the same, that is, the change of the n\irabeia does not imply any othei change If. for rJiample, the increased number of men should be in each other's way, so a» to interfere with their Uboui, tUij inujt be made a feiMralr contideration. It 1-1 INTRODUCTION. (2.) General Method: 13 ; ijd. :: 98 I d. req. = 175 x 98+ 13 =• i3-i dayt. (3.) By Ratio : Here ij (diys) is to be increased in the ratio of 98 to 13. r75X98-M3 = i3-2. Kx 2 If 13 men make 420 yards in 20 days ; how much will they make in 11 days? Note. — The number of men remaining the same, while the time and tlie *rort change, need not be noticed. (1.) 420 yds. in 20 d. is 21 yds. in i d., and 11x21, or 231 yds. in 11 days. (2.) 420 yds. ; 20 d. :: yds. req. : 11 yds. req. = 11 x 420-=-20 = 23i yds. (3.) Here 420 is to be diminished in the ratio of 1 1 to 20. Bx. 3. A pump, A, delivers 1 ton in 5"'; another, B, i ton in 8"; and a third, C, I is 15'" : how mucli water will they dtUver in 1'' iC" .' Ans. A, '»=i4tou6; B, 'j° = 87; C, ^ = 47. Total, 27-4 tons. Es 4. A boat. A, lauds 52 men in 28"' (going and returning) ; another, B. lands 68 mej in 41'" ; aud a third, C, lands 20 men in 23"'; how long will all take to land 220 At these rates, in 1'', A lands ^g x 52 men = 111-4; B, 5? x 68, = 99-5 ; and C, <§ x 20, = 52-2. Total m i'', 263-1 men. Now, as the number landed is proportionato to the time, we have 263-1 : 1" :: 220 ; 220 x i -r- 263-1, or o''-84 nearly. Ei 5. A boat, A, fills 8 tons of water in 34'' -, another, B, fills 5 tons in 4" ; and ■ third, C, fills i.J ton in if" ; in what time will they fill 107 tons ? (1.) In i", A fills ^ tons; B, J tons; and C, f tons; or altogether, ij=/ tons. This it I too in fj»3 of i", 107 tons in 28 x 107-=- 123 = 24l'-4. (3.) Having found the fraction expressing the joint effect for i"", or <;/ tons ; i^ is to he changed in tliat ratio, whicli will convert this into i, (ff^ by" E,v.), which gives the time for i ton ; this is then to be increased in the ratio of 107 : i. Note. — Such questions as in Ex. 4 and 5 do not usually admit of exact solution ; thus, in sny whole number of trips that can be proposed, the boats carry too much or too little. Each boat performs a certain quantity in one particular interval of time, and not continuouely, like a pump, or so much per hour : the reduction, therefore, to hourly rate, is not correct, but it is near enough for forming a tolerable estimate, which, in practice, is all that is wanted* To obtain as complete a solution as the question allows, we must take each boat's performance separately, and add them all up. Ex.6. The change of the sun's declination in i day is 1 8' 2 1" ; find the change for i" 34". 24" (1440"') : j8' 2i" ("oi") :: i*" 34"' (gi"') : x or, less exactly, 24'' '. i8'-3 :: i''-6 ; x. Ex. 8. Against ii" in a Table standg 6726, and against ii° 30' stands 6354 ; find the term corresponding to 11° 37'. 30 : 372 :: 37: X to be subtracted from 6726, which givi term required. Ex. 7. In a Table, against 36° stands the term 27943, and against 37° stands 28504; find the term corresponding to 36° 23'. 36° ^7943 37 ^8504 Diff 561 Hence 60 ; 561 :: 23 . jr which added to 27943 (because the terms increase while the argument* increases), gives the term required. 50. The process of finding a term which falls between two given terms, or, hs it is callsd, Interpolation, is sufficiently exemplified above ; but it is important to remark that it w not always necessary to work proportions at length. It is enough, for most practical • The argument is tne quantity at the side or head of the Table, for which the terms 01 :(uuititie£ in the bod; of the table are given. PROPORTION. 16 jiuri)yses, lo take a quantity, somewhere between the given terms, as half waj, or a third of the way, between them, according to tlie case. Tlie power of guessing tlie proportional part is acquired by practice, and saves time whicii otherwise would often be wasted in working to a superfluous degree of accuracy. On tlie otiier hand, when extreme precision is required, this pro- portioning alone is not enongii, but a correction is necessary, for which see the e.xplanation of the Tal)le for finding the Equation ol Second Ditferences. [2.] Double Rule of Threi, Direct. 01. Questions in the Rule of Three occur also in a more complex form ; thus, if 2 men do 7 yards of work in 3 hours, liow many yards will 13 men do in 11 hours? in which the answer is required to correspond not merely to a certain number of men, but also to a certain number of hours. This (juestion resolves itself into two: 1st, if 2 men do 7 yds. how many will 13 men do in the same time, or 3 hours? The answer to which is 45-6 yds. ; and, 2nd, if 13 men do Abh yds. in 3 hours, how many yds. will tiiey do in 1 1 hours? Hence the solution of such questions is called the Double, or Com])ound Rule of Three. Ex 1 . The example above (1.) I man does i of 7 yds., or 3-5 yds. in j"", and 13 men do 45-5 yds. 13 men do 45-5 yds. in 3'', or \s^i^ yds. in 1'', and tlierefore 166-87 in 11 houi-s. (2.) Tlie two statements as given above. (3.) 7 is to be increasi,d in the ratio of 13 ; 2, and tlien of 11 ; 3, Ei 2. If 9 men make 47 yds. in 4 days, how many yards will 17 men make in 31 days.* Ans. bti yds. El. 3. If 5 men do 64 yds. in 1 1 days, in how many days will 14 men do 37 yds. ? (1.) I man does 64 yds. in 55 days, or i yd. in c-86 days, and 14 men do i yd. in -86.4-14, a"J 37 yds. in 2-27 days. (2.) 5 m. : 64 yds. :: 14 m. : 179-2 yds. 179-2 : 11 :: 37 : 2-27 nearly. (3.) 11 is to be diminished in the ratio of 37 : 64, and then of 5 ; 14. »!». 4. A certjiin quantity of provisions lasts 170 men for 3 months; how much Is re- quired for 210 men for 2 months ? (2) 170 : I (whole) :: 210 : x = 210^-170. And .v : 2IO-S-I70 :: 3 : 1. (3.) The quantity is to be increased in the ratio of 210 '. 170, and diminished in the r«tio of 2 : 3. Kx. 5. A steam. vessel has fuel for steaming 13 days at 11 hours a-day ; how much m-a«t she take to steam 15 days at 18 hours a-day.' (3.) The fuel must be increased in the ratio of 15 : 13, and then of 18 : 1 1. {J « H "" {\% which is i|U, or i\\ nearly, or nearly doubled. Ex. G. Three boats fill 16 tons of water in 7 hours; how many boats, at the mau average performance, will fill 78 tons in 10 hours ? (1.) 3 b.,Hts fill i6tonsin7\ or ^of 16 = 1-3 tons in i\ and 23 tons in 10 hours. Thea. since 23 tons employs 3 boats, i ton employs ^ of 1 boat, and 78 tons will employ or lo-i boats. '2.) 7' : 16 1. :: 10° : x tons (--22-9' 22-<) t. :3 b.:: 78 1. : 102 h. (;j.) ] IS to be Increased in the ratio of 16; 78, and then liminishcd in the ritio of 10 ; ». 16 INTROUUCTIOH. 2. Inverse Proportion. 62. In direct proportion, as we have seen, more is always ftiJ- lowed by more, and less by less. But when the nature of the qiies*- tion is evidently such that more will be followed by less, or less by more, the proportion is no longer direct. For example, if 5 men do certain work in 4 days, in how many days will 7 men do the same work? Here it is evident that the greater number of men will require less than 4 days. Again, if a ship going 8 knots, sails a certain distance in 5 hours, it is evident that, if she goes at a. greater rate, she will perform the same distance in less than 5 hours. 53. In a question of work performed, the result is represented by the number of agents mnltii)lied by the time each works; tims, 6x5 or 30, represents the labour of 6 agents working for 5 hours, the unii of work being that performed by 1 man. If now, the work remain- ing the same, we double the number of agents, we shall obviously halve the time, since 12 men will do the work of 6 in half the time, and 12x2^=30. Or, again, trebling the number of agents, gives 18 X ^ of 5, or 18 x i=30. That is, while one factor of a given pro- duct is increased in the ratio of 3 : 1, the other must be diminished ia tlie ratio of 1 : 3, which last ratio contains the same terms as the other, but in a reverse or inverted onler. The four numbers consti- tuting two equal prodixcts are hence said to be in inveise proportion to each other In the example. No. 52, the number of men is increased in the ratio of 7 : 5, and the time is accordingly to be diminished in the ratio of 5:7; hence 4 days becomes 4 x 4, or 2^ days. [1]. Rule of Three Inverse. 54. In regard to the solution of these questions ; (1.) In the method of unity, the consideration of inversion does not present itself. (2.) As a question of proportion, the solution may be effected thus. Suppose the proportion were direct, then (example above, keeping the antecedents and consequents in their given order) 5 men : 4 days : : 7 men : x days. Now, we require a direct comparison between the number of men in the two cases, and the times in the 'wo cases ; hence we alter this to 5 men : 7 men : : 4 days : x days. But this would give x greater than 4, as 7 is greater than 5, whereas we know it must be less ; hence, inverting the last two terms, gives 6 : 1:: X -.A, or 7 : 5 : : 4 : x =—-—=—, or 2^ days. Hence the pro* i*ess (which is, perhaps, as little liable to mistake as may be expected m a question of some perplexity), is, 1, to write, in the form of a direct proportion, the given antecedents and their consequents ; 2, to close terms of like denomination ; 3, to invert the last two terms, and then to find the unknown term. E«. I. If 7 men do ccilmn work in 4 days, in liow many days will 10 men do it ? ( 1.) 7 men in 4 days ii i man in iS days, and 10 men in z-% days. PROPORTION. i I [%) Diroct form, 7 men : 1. d. :: 10 men ; days required. Like terms, 7 : 10 :: 4 '. days required. Inverting, 7 : 10 : : d. req. ; 4. Ans. ;. 8 days (.3 ) It is evident that 4 is to be diminished in the ratio of 7 to 10. ti 2. If 17 men do certain work in 14 days, how many men will do the same work in 4 days? (1.) 27 men in 14 days, is i man in 378 days ; and 378 -!- 4 gives 94J men. (2.) Direct form, 27 m. ". 14 d. :: men req. : 4 d. Closing like terms and inverting, men req. = 17 x I44-4»94{ men. (3.) 27 is to be increased in the ratio of 14 I 4. t' . 3. If 12 men do certain work, working 4 hours a-day ; how many men will it take to do the same work, working 7 hours a-day ? (I.) 12 men in 4 h. is 48 men in i h., and V in 7 hours, or 7 men nearly. ^2.) Direct form. 12 m. I 4 h. t: men required ', 7 h. Closing like terms and inverting, 12 x 4-5-7 = 7 men nearly. (3.) 12 is to be diminished in the ratio of 4 ; 7. Bt. 4. Certain tons of fuel last a steam-vessel 1 1 days, steaming 4 hours a-day ; how lonj will they last steaming 61 hours a-day .' (1.) 4 h. for 1 1 d is at the rate of i h. a-day for 44 d., and therefore 6} h. for 44 ->-6J, or 88-=- 13, which is 6-77 d., or 6 d. 185 h. (2.) Direct form, 11 d. '. 4 h. r. x dayi : 6.{ h. Closing like terms and inverting, a' = 44-!-6S = 6-7; d. (3.) Here 11 days is to be diminished in the ratio of 4 to 6J. Ej. 5. A certain quantity of fuel lasts a steam-vesael 12 days, steaming day and night ; how long will it last steaming 14 hours a-daj .' Ans. 20) days. Kt. 6. A pump, A, empties a cistern in 3 hours; another, B, in 2{ hours ; in what time will they empty it both working together ? (1.) In i'', A empties J of it, and B empties i-!-2j, or i^J, which is J. Hence in i" both together empty J + j, or {. Suppose, for greater convenience, the cistern to hold 10 tons ; then in i*" both empty '," tons, or i ton in i''-f-'|J', or i"" x ft, =ft ol i"", which is 10 tons in ^ of i'', or i;;*'. (8.) Stating the question directly, we should say, J-f J ( = J) ; the whole, or i : : time required : i\ But, the greater the fraction representing the hourly work done, the smaller must be the time required for any given quantity of work. Hence I ', J :: 1^ '. time required =^ of i''. (S.) Here i*", in which the fraction { is done, is obviously to be increased in that ratio which will turn { into i, or the whole ; and this ratio is f, for | x {= i. Ki. 7. A can do certain work in S*', and B the same work in 6* ( in what time will they both complete it together ? (1.) In i'' A does j, and B J, hence both together J-hJ, or ^. Let the work be repre. sented by 10, then in 1'' both do JJ, and tlierefore they do the unit of work in l"" + JJ, or ^ of 1". Hence they do the whole in 10 x ^ = y of i*', or 3^ . (2.) Direct form, ! + J I 1 (whole) :: time required : i' = V- {3. ) i"" is to be increased in the ratio of 24 : 7. Si. 8. Five pnmps empty a cistern in 1 3 hours ; how many must be put on to empty it is 3 J hours ? (1.) I pump in 65 hours gives i8-6 in 3I hours. (2.) 5p. : ij"" :: j:: 34". Ultimately, *-sx 3-^3-5, (3.) 5 is to be increased iu the ratio of 13 ; 3 J. Ki 9 Four pumps empty a cistern in 10 hoon ; how long will 7 such pumps t«koT Ans. 40 + 7=^ St>7. INTRODUCTION. 10. A certain .juantity of bread lasts i lo men zi days ; how long will it laai 74 men? (1.) 21 d. for no men is i d. for 2310 men, and 23io-i-74 gives 31-2 dajs. (2.) Direct form, no m. : 21 d. :: 74 m. : :r d. Closing lilie terms and inverting, a-=2i x iio+74=3i-2 days. (3.) It is evident tliat 21 is to be increased in tlie ratio of 1 10 ; 74. days at four-fifths allowance ; how long (1.) J lasts 21 d., J will last 4 X 21 or 84 days, and f, or whole allowance, ',♦ or i6-8 days. Hence J allowance will last 3 x i6-8 d., or 504 d., ana J, one half of this, or 25-2 days. (2.) J : 21 :: 5 :: required days. Closing and inverting, days required <=2i "1-^1 = 2$ 2 days. (3.) 21 days are to be increased in the ratio of J ; |, that is 21 x }-e|. Ex. 12. If it takes 54 yds. at J of a yard wide, to cover a surface ; how many yards will it take at J of a yard in width ? (1.) H yds. at I wide is 3 x 54, or 162 yds. at ^ wide, or 40-5 yds. at i yd. wida. This is 5 X 40- 5 or 202-5 y^s- at J wide, and J of this, or 5062 yds. at j vride. (2.) Direct form, 54 yds. ', f width i; yds. req. I |. Closing like terms and inverting, yds. req. = 54 x |^|=s 50*62 yds. (3. J Here 54 is to be diminished in the ratio of f : J, or of 15 ; 16. [2.] Double Rule of Three, Inverts. 55. As tlie inversion arises from a product remaining constant while both factors vary, questions of this kind may be solved directly by taking, in each of the two proportions necessary, those terms only which are directly ]iroportional to each other. For example, in a question of agents, work, and time, the first proportion would include work and time, and the second, agents and work. 111. Logarithms. 56. These are numbers calculated for the purpose of converting multiplication into addition, and division into subtraction. 1 . Use of Logarithms. 67. Every logarithm consists of two parts, the index and the decimal part;* thus, in the logarithm 2-80618, the index is 2, and tiie decimal part -80618. 68. To find the Logarithm of a given number. Find in the Table of Logarithms of Numbers the decimal part (for which see also the Ex])lanation of that table); and then api)ly the index by one of the two following rules: — (1 .) When the number consists of a whole number, with or with- out decimals, the index is 1 less than the number of fi^vm'es in the whole number. This part is also :8llcd the mantissa. LOGAIUTHMS. ] 9 Ex. 1. Find the log. of 52a. Against 512, iu the Table, stands •717671; then, sime there aie three figures in $". th» indei is 2; hence the log. is 2-717671, Ex. 2. Find the log. of 5-22. The log. of 5-22 ia o-7i767i, because there is one figure in the whole number, and ou< less than i is o. (2.) When the number consists of decimals only, count the number of ciphers between the decimal point and the first significant* figure after it, and subtract this number from 9 ; the remainder is the index. Ex. 1. Find the log. of '0058 14. The decimal part of 5814 is -764475 ; there are two ciphers before the 5, which 2 taken from 9 leaves the index 7 : hence the log. is 7-764475. Ex. 2. The log. of -5814 is 9-764475, for the number of ciphers before the -5 is notMufi which leaves 9 for the index. 59. To find the natural number of a given Logarithm. Look for the decimal part of tlie given log. in the body of the table, and take out the number from the side column and top. To place the decimal point. Add 1 to the given index of the log., and mark ofi' to the left this number of figures ; these will be whole numbers; the rest, if any, will be decimals. If tlie index is 9, put the dot before the first figure; if it is 8, prefix one cipher to the first figure, and place the dot laefore the cipher ; if it is 7, prefix two ciphers, and so on.f Kx. 1. Find the number to the log. 1-717671. The number (to 4 places') to -717671 is 5220 : adding i to the index i, gives 2, which, marked otf to tlie left, gives 52-2, the number required. 2i. 2. Find the number to the log. 8-581381. The number to 581381 is 3814; prefixing one cipher gives -03814, the number required. \Vhen the number exceeds four figures, see the explanation of llie table. 60. In using logarithms, it is proper to observe that the number (whether it contain decimals or not), and the decimal part of the logarithm, are in general true to the same number of figures, re- jecting prefixed ciphers ; thus, for instance, the log. 3-7575 corre- sponds to the number 57-21, and the log. ;3-7576 to 5722, nearly. So also, 8-7575 to -05721, and 8-7576 to -05722. This remark should be kept in view, because it is mere waste of time to employ more figures than are required to insure a certain degree of precision in the result. • That is, the first figure not a cipher. t As the index of the log. is 1 less th.in the number of figures in the natural number Itaelf, it would follow that the index of -3814 (for example) in which there are no significant figures, would be 1 less than nothing, the meaning of which is, that such a log. is reckoned on the opposite direction from a certain point, which need not here be further discussed. The index of such a log. is called negative .- ana as this is cmbarriLssing to beginners, 10 is added to the index 0, whi-reliy 1 less gives 9. But 9 is the index, properly, of a number consisting of 10 figures; however, as we have no such numbers to deal with, the ambiguity of thii Jnabli ununing is not exuerienced. 20 INTIIODUCTION. 61. The remark (No. 3.5) applies also to logaritlims -, thus, for example, if we propose to use only four figures of the log. -88 1385, we write "8814, which is evidently nearer to 'SSlSSo than •8Slo would be. Again, if we take four figures of •88134.3, we write '881 3. 62. To find the arithmetical complemeyit of any number or logarithm. Take every figure from 9, except the last, which take from 10. It is necessary to begin at the left. Kx. I. Find the arith. comp. of i'87045 | Ex.2. Find the arith. comp. of o'gijso arith. comp. log. required 8-12957 | arith. comp. log. 9-08650 63. A subtractive quantity is, by this means, made additive The process is equivalent to subtracting the number from 10, and the reason of it is evident on considering that to add 3, for example, and subtract 10, is the same as to subtract 7. In like manner, in- stead of subtracting 47"" 32% for example, we may add Vl"' 28' (the complement to 60), provided we subtract 1 hour (or 60); and thus any number of quantities, of which some are additive and some sub- tiactive, may be rendered all additive, provided that the larger numbers which are employed in taking the complements be tliem- (elves subtracted. 2. Certain Arithmetical Operations by Logarithms. [1.] MuUiplieation. 64. To multiply numbers together, add their logarithms together; the sum is the logarithm of the product required. Ei. 1. Multiply 530-9 by 27-22. 530-9 log. 2-725013 27-22 log. 1-434888 /Im. 14451. log. 4-159901 Ex. 2. Multiply -079 by 3-142. -079 log. 8-897627 3-142 log. 0-497206 .\ns. 0-2482 log. 9-394833 [2.] Dirhion. 65. From the log. of the dividend subtract thelog. of thedivioor; the remainder is the log. of the quotient required. If the logarithm of tlie dividend is the lesser of the two, increase its index by 10. •ii. 1. Divide 4280 by 365. Ex. 2. Divide 69-3 by 71-7. 4280 log. 3-631444 69-3 log. (+ 10) 1-840733 365 log. 2-562293 71-7 log. 1-855519 Am. 11-73 log. 1-069151 I Ans. 0-9665. log. 9-985214 [3.] Involution. 66. Involution is the process of multiplying a quantity by itself: the quantity thus multiplied is said to be raised to a power. 67. The first power is the number itself. The second power is the number multiplied by itself; this is also called the square. TSe third power is the number again multiplied by itself; this is also p«licd the cube. PRACTICAL GEOMETllY. 21 The number or quantity to be raised to a power is called the root; the number which indicates the power to wliich the quantity is raised is called the index. 68. To s/fuare a number. Multiply the log. of the number by 2; the product is the log. of the number required. When the number is a decimal fraction, subtract the index (after being doubled) from 10 multiplied by 2 (or 20), diminish the re- mainder by 1, and prefi.x the number of ciphers indicated by this remainder to the number corresponding to tiie logarithm. Ex. 1. Square 12-39. 12-39 log. i*09307i Ans. I53"5. log. 2M86142 Ex 2. Square "0592. •0592 log. 8-7723» Ans. -003505 log. 17-54464 17 from 20 leaves 3 ; deducting i gives » | 2 ciphers are, therefore, prefixed to 3505. 69. To cube a number. Proceed by tlie above rule, only reading 3 for 2, and 30 for 20. In like manner, to raise a number to the fcmrtA power, read 4 for 2, and 40 for 20, and so on. [4.] Svolulion. 70. Evolution is tiie reverse of involution, and is the process of finding that number which, multiplied by itself a certain number of times, will produce the given number. This number is called the root of the given number. 71. To extract tiie square root of a number. Divide the log. of the given number by 2, the quotient is the log. of the square root required. When the given number is a decimal fraction (that is, when the index of its logarithm is 9, 8, 7, &c.), increase the index by 10. I. 1. Find the square root of 1-53 '■S35 log. o-i86ic8 2 )0-186108 ij9 Sq. root req. 0-093054 Ex. 2. Find the square root of -003505. -003505 log. 7-54469 1 )17-54469 Sq. root req. 8-77234 72. To extract the aibe root. Proceed by the above rule, only reading 3 for 2, and 20 for 10. To extract the fourth root, read 4 for 2, and 30 for 10, and so on for other roots. IV. Practical Geomethy. 1. Definitions, 73. Geometry is the name of that science which relates to the measures of space. A PHOBLKM is something required to be done. 'VI INTRODUCTION. Parallel lines are lines so placed that the shortest distance between them is every where the same, as A B, C L>. Such lines evidently never meet. 74. A CIRCLE is a figure bounded by a curve line called the di- cuviference* of which every point is at tlie same distance from a point vvithin, called the centre. Thus, A B D is a circle, and C the centre. 76. The circumference is divided into 360 equal parts, called degrees, written thus, 360° ; each degree, into sixty equal parts, called minutes (60') ; each minute into sixty seconds (60") ; and also each second, into sixty thirds (60"'). Example, 11° 19' 46", eleven de- grees, nineteen minutes, forty-six seconds. 76. The circumference is also divided into 32 equal portions of 11° 15' each, called points of the compass. These are again subdi- vided into half points and quarter-points. The term point is used indifferently for the arc of 11° 15', and for a mere point of division of the circumference. 77. A straight line, A B, drawn through the centre, divides the figure into two equal parts, called semicircles, as A D B, A E B. The half circumference measures 180°. 78. Ihe line A B is called the diameter: it is evidently equal to twice tlie distance from the centre, C A, which is called the radius. language, circl« and circumferencp i oiicle is properly thp tur/ace or nrea oi ! often used indifferently tlie one fiv c f.^ure iuL'tuded within tlie ciicum PRACTICAL QEOMETRT. 23 79. If anotlier diameter, D E, cross this, and divide each semi- circle into two equal parts, the four equal parts, A U, B D, B E, E A, are called quadrants, and each of such portions of the circum- ference measures 90°. 80. Any portion of the circumference is called an arc, and the line joining its extremes is called a chord: thus the line B F is the chord of tiie arc B F. 81. An ANGLE is the inclination of two straight lines to each other; that is, the difference of the directions in which they lie: thus A B C, or B, is the angle contained by the two lines B A, B which are called the legs. An angle is not changed by increasing or diminishing the length of the legs, because the length of these lines has nothing to do with the directions in which they lie. 82. Since in describing a circle the radius moves round the centre C, exactly as the point of the compasses advances on the circumfe- rence, the angle A C B is measured by the number of degrees in the arc A B. 83. The arc A B is said to subtend the angle A C B. 84. An angle of 90°, as A C D (fig. in No. 77), which is anb- tended by a quadrant, as A D, is called a right angle. A circle con- tains four right angles, a semicircle two. 8.5. The angles A C D, BCD, being each 90', are equal ; and C D, which makes these adjacent angles equal, is said to be perpen- dicular to A B. 86. The difference between an angle and 90° is called its comple- ment ; the difference to 180° is called its v/ppkment. 24 INTRODDCTION- A.I1 angle less than 90° is called acute, as A. An angle greater than 90° is called obtuse, as 87. A PLANE TRIANGLE IS 3 figure Contained by three straight lines. When the three sides are equal, the triangle is called equilateral; when two of them are equal, it is called isosceles. 88. When one of the angles is 90°, the tr'.angle is said to he ric/ht - anyled; when each angle is less than 90°, it is said to be acute-angled; when one is greater than 90°, it is said to be obtuse-angled. Triangles that are not right-angled are called in general oblique- angled. 89. In a right-angled triangle, as A BC, the side AC, opposite the right angle is called the hypothenuse ; one of the other sides, as B C. is called the base ; and the third side, A B, the perpendicular. 90. A SPHERE, or GLOUE, is a solid figure bounded by a curve lurface, of which every point is at an equal distance from the centre. 2. Geometrical Problems. 91. The instruments necessary in constructing tlie figures in these problems are, a pair of compasses and a straight edge of any kind, as of a ruler, or, when such cannot be had, the back of the fold made by doubling a piece of thick paper. Also the parallel rulers are convenient. These may be of the common form, which needs no description here, or those called Marquoi's Rulers.* 92. The accuracy of a straight edge is tested thus. Draw a line with a fine pointed pencil, or steel pen, along the edge, between twc points near the extremities. Then turn the ruler over and draw another between the same two points : if the edge is perfect, the two jnes will appear as one; if not, there will be a space between them. • These last consist of a right-angled triangle, having one of its angles about 20°, and a flat ruler somewhat longer than the hypothenuse of the triangle, both of the same thickness. By sUding the triangle along the edge of the ruler, which is kept fixed, two sides of it move parallel to tliemselves. This parallel motion is perfect, which is not always tiie rase with tha ooromon parallel rulers, especially after long use ; and besides this, the triangle being light- angled, dispenses rith the trouble of drawinj; perpendiculars bj points. PRACTICAL GEOMETKY. 25 93. Problem. To draw a line tlirougli a given point parallel to anotlier line. C is the given point, A B is the line. Take the shortest distance from C to A B in the compasses ; set one foot on A B as at B, and describe a small arc ; then tlic line drHwn through C, so as to touch this arc, is the line required 94. Problem. To draw a line [)arallei to another line at a given distance from it. A B is the line, C D the given distance. Take C D in the compasses, place one foot near each end of A B, and describe two ares; the line drawn touching tiiese ares is the line rcfpiired. 7==^ !J5. Phublkm At a given point in a line to make an angle equal to a given angle. P is the point in the line PQ; A is the given angle. From the centre A, with any convenient radius (the longer the more accurate), describe an arc, C B ; from the centre P, with the saran radius, A B, describe an arc, D E ; take the distance from C to B in the com- pas:»es, and put one foot on D and the otiier on the arc at F, and join P F : tlien the angle F P D is equal to B A C, tlieir measures, V D and B C, being the same. 96. Problem. From a jmint M, in a straight line A M,to draw a perpendicular to it (fig. p. •2(i). (1 .) Draw a straiglit line any where, and set off by the compasses 5 equal parts upon it. With 3 of these parts in the compasses, as radius, describe from M, as a ceiitre, an arc at I ; then lay off 4 parts from M to A ; with 5 paits, as radius, describe from the centre .\ an »rc cutting the former arc at I ; join I M : this is the perpendicular required. INTRODUCTam. 1 he following methods are also used : (2.) When the point M is at or near the end of the line. Take a point P, such that a line supposed to join P and M may make the angle P M A about 45° ; and from P as a centre, with the radius P M, describe a small arc I, and another opposite, as K, draw the line I P K, and join the point where it crosses the arc K with M. K M is the perpendicular required. (3 ) A^'hen the point M is not near the end of the line. Take two points P, B, at equal distances, from M, and at P and l as centres with a radius exceeding- P M, describe two arcs, cutting each other at I ; join I M. This line is the perpendicular required. A. 97. Problem. From a given point without the line, as C, to draw a perpendicular to it. (1.) When the point is nearly opposite the middle of tlie line. Take in the compasses a distance exceeding the distance from C to the line ; and from C, as a centre, describe an arc, D E ; then, fron. 1) and E as centres, with a convenient radius, describe two arci cutting each other at I. CI is the perpendicular required. PRACnCAIi UEOMKTRV. 27 (2.) Wlien tlie given point, is towards tJie entl of tlie line. Take a point P"as centre, and with P C as radius describe an arc C D. Take anotlier point Q as centre, and witli Q C as radius describe anotlier arc cutting C D in 1. CI is tlie perpenaicular required. 98. Proislem To bistct a line A B, or to divide it into two equal parts. Take in the compasses a distance e.xccedinjj half the line, and from A and B, as centres, describe two arcs. Tlie line I K, joining the i)oints of their intersection, divides the lino A B into two equal parts, A M, M B. / M ■ \ / 99. Prodlem. To divide a line, A B, into any proposed number of enuiil parts, as fiv'o, for example. Draw a line A C, making about half a riirlit angle with A B. Draw- another line, B D parallel )o A C. On A C and B D lay otf INTROUUCTIOX. riv»; C(jual parts; join the points 1 ami 4, 2 and 3, &c. ; these iinct «iil divide A B into 5 equal 'arts. / A. \ In like manner, the line might be divided into any other rannbcr of equal parts. • 00. Problem. To bisect an arc A B, or an angle A fj B. From the points A and B, as centres, with a radius exceeding half the distance A B, describe two arcs cutting each other in I, and draw the line C I ; C I bisects the arc A B, and the angle A C B. If the angle alone is given, and not the arc subtending it, describe this arc from C as a centre, with any convenient radius. 101. Problem. To divide a circle into 2, 4, 8, &c. equal parts. Draw the diameter A B ; this divides the circle into two equal parts. From A and B, as centres, with a radius exceeding half A B, describe the arcs I and K, cutting each other above and below A B ; join I K : the line E D is a diameter crossing A B at right angles, and dividing t'.ie circle into the four quadrants, A E, E B, B D, and D A. Bisect the arc A D (No. 100); draw the diameter through C: this will bisect B E also. Bisect, in like manner, B D and A E. The circle is now divided into 8 equal parts, of 4 points each ; bisecting these last arcs divides the circle into 16 f qual parts, of 22°J each • PRACTICAL UKOMKTRV. dnd ugahi bisecting these dividts it into ttie 32 points of tlie comiiass o/Jl"' <;ach. All arc is divided into a number of parts not divisible by 2, as into .3, 5, 7, &c. parts, by trial. 102. PuoBLEM. To find the centre of a circle, or circular arc. Take two points, as A B, on tlie cirjiimference, and join them; bisect the line A B (No. 98), and at the middle point draw a per- rendicular (No. 96, 3d). Take a third point, D, join it with B; bisect the line B D, and draw a perpendicular at the middle Ihc two perpendiculars will cross at the centre. point. 10-'!. Pk(;ulem. To draw a circle throuuli three i^^iven jioints. iSiipposi; the tiiree points to lie in a circle, and proceed to tind tbr centre as above. It is easy to see that hovvever three points may be placed, some ciie circle will always pass through them; for an infinite number of circles may be drawn passing through two i)oints, and therefore some one of these must likewise pass through a third ])oint wherever situated. 3. ("strand ( 'oiinlriirtMU of the Scales. 104. These are flat, thin pieces of brass, ivory, or wood divided in'.o certain portions by lines, and serve for measiiniij^ or laying oil lir eu or disin id ait/jles 30 TNTROnUCTIOTf. Tho coiiiinoi. scale of equal parts lias genera'ly on ono side fnur or fi\rf tlirterent scales for cliH'erent measures, on each side of wliic!; one division is subdivided into 10 equal parts. 105. In the diagonal scale, the shorter lines dividing the length into equal portions (units) are crossed perpendicularly by 10 others extending the length of the scale. The end division, or unit, has its upper and lower edge subdivided into 10 equal parts, and diagonal lilies are drawn from the beginning of one division to the end of the opposite one. This ettects a further subdivision by 10, as an example will shew. To take the No. 5-28 from this scale by the compasses. Set one foot at .5, and the other at the second line on the lower vi['j;e of the subdivided unit, — this gives 6-2. "Now follow np the diagonal line at the -2 to the eighth of the long parallel lines, and, fixing the ]toint there, extend the other point to meet the line which rise-! at 5, crossing the breadth ; and the number Is taken. The same process serves for tens and units, as for units and tenths, and so on ; thus the No. 52-8, or 528, is taken as above. By placing the points of the compasses between, instead of oh, the 10 louir parallel lines, we may obtain a still further subdivision. >; ■.T-^-.^^^N^^>jv^,^-^.^^^v^^^^^^^ Diagonal -ScaL.. ■'/•"/"-■■'"/.^■■^■■■■^■■j>^-''-j:.^ -1 8 -« 1 1 H -: # B = = = = = = ^ = 2 i i = ^ ^ r ^ ^ = 5 B ' lOG. Angles are measured, or laid oH', either by means of the lines marking the divisions of degrees, or half degrees, at the edge of the scale, and which are numbered at each 10° or 5°, or by means of the Scale of Chords. (1.) To measure an angle by the marked divisions. Place the middle point of the scale (which is strongly marked) upon tlio angular point, and lay the edge along one of the legs; the other leg, produced, if necessary, shews, on the graduated edge, the degrees which the angle contains. (2.) To measure an angle by the scale of chords. Take in the comj)asses the chord of 60° otf the scale, and describe an arc: taki« tlu! distamte from A to B in the compasses, and, placing one foot at PRACTrCAL GKOMETKr. ?,l tli« boj^inning of the scale of chords, look liow nianj Jegroos the other foot extends to. Thus, for example, if A B extends to 27°, the arc A B, or angle, C, contains 27°. 107. To lay off an angle from a given line, as, for example, 27°, Describe an arc AB (fig. lOG), with the chord of 60°, from C, as centre, and set off the chord of 27° from A on A B ; join C B, and A C B is the angle required. When the angle to be measured or laid off exceeds 90°, measure or lay off 90°, and then the excess above 90°, 108. Tiie semicircle with a graduated edge is useful for this p'.ir|)ose ; hut the most convenient instrument, especially for using witli tlie chart, is a transparent horn semicircle, with a long silk thread attached to its centre.* 10!). To construct a scale of chords to any proposed radius. The radius is equal to the chord of ()()°; describe, therefore, a quadrant, divide it into portions of 30°, 20°, 10°, and so on; draw tlie several chords, and transfer them to the proposed scale. 4. Tlie Sector. 110. The Sector is a ruler, or scale, which folds into half iti length by moving round a large circular joint on which it is accu- rately centered. Several lines, or scales, are laid off from the centre to the extremity on both legs of the sector, as tangents, sines, &c., and others parallel to the edges. We shall refer here only to thai one which is called the line of lines (marked C L in the figure), on account of the great convenience of the sector in reducing a plan, or a figure, to another on a ditferent scale, dividing lines propor- tionally,t and in solving some simple questions which dej)end on |iroj)ortion alone. The line of lines is divided into 10 equal jiarts, and these again Hre siuiilarly subdivided. The distance from the centre to any point in tlie line of lines is called the lateral distance; and that between any point in the line of lines on one leg, and the correspoiuling point on the other, the trnnsverxe distance. • Sucti einiieirc'les, made of horn or other transparent material, and h.iviiiga long silk tliread attached to the centre to extend a straight line to any point beyond the circnmlVr- ence, are must useful, especially for chart work. They are commonly called protractors. t Another instrument, equally conveuient and portable, but more expensive, is tho fiTopurtional cvmiiasses. These C(jmpa6Sis open on a movable centre, so that the opening of one pair of points may, by moving the centre; l>« made to bear any proportion to th« opi.uug of the other pair of point«. o2 INTRODUCTION. The following examjales will illustrate the use of the Sector.* Ex. 1. To divide a lin« into a number of equal paits, as for ex. 7. Take the given line in the compasses; place one point on the division 7 on one leg of the sector, and open it till the other falls on the other 7. Then the transverse distance i to I is i-7th, 2 to 2 is 2-7ths, and so on j or the line 7, 7 is equally divided into the pans I, I ; 2, 2 ; &c. Ex. 2. To reduce a plan on the scale of 3 inches to a mile, to another scale of 2 inches Take the lateral distances on tlie scale of the 3-inch plan. Take 2 in the comp;isses j place one point at the division 3, and open the sector lill the other poini falls on the other 3. Then the transverse distances will be the distances on the pronostd plan. Ex. 3. A line of a given figure measures 85 ; find the measure of another line in the same figure. Take the given line 85 in the compasses and open the sector till their points measure me irnnsverse distance 85, 8-5. Then any other line of the figure taken in the compasses is mea-ured by finding the Cirre-ponding points in the two legs ■which exactly contain it, and multiplying the number shewn by la • See J. F. Heather on Mathematical Instruments, Lockwood & Co., Ludgata Hill. V. Raising the TfiiGONOMETaicAL Canon 111. Thi-i term implies forming the proportions or analogies pro- per for the solution of problems concerning right-angled triangles. Before, h&wcvcr, the student proceeds to the actual composition of these analogies, lie should be acquainted with the few propositions of geometry whicli are given in the following section. 112. Definition. An Axiom is a proposition assumed to be so obvious as to require no demonstration. The principul axioms whidi have been employed as the founda- tion of geouietrical reasoning are the following : — (1.) Geometrical magnitudes are said to be equal when one being placed on another coincides with, or exactly covers, it. (2.) Two magnitudes which are each equal to a third, are equal to each other. (^3.) If equals be added to equals, the wholes will be equal. That is, if two magnitudes be equal, and a third be added to e-jch, the two sums will b< (4.) If equals be taken from equals, the remainders will be equal. (5.) If the same or equal quantities be added to unequals, tiie Bums will be unequal. (ti.) If equals be taken from unequal^, the remainders will be unequal. (7.) The halves of equal things are equal. (8.) The doubles of equal thing.s are equal. 1 13. Def. a Geometrical Theorem is a ]ir(>uositi(>n in wliich some property of a figure is demonstrated. The term Proposition includes both Piv,i)lems and Theorems. 114. Df.f. a Corot.i.auv is an obvious Cviucliisioii oi' neee>»sttr;y inference, from a proposition. I. Theorems of (hwnclry. 1 15. A straight line, as A C, standing on anotiier, as 1) E, mokfis the adjacent angles, ACE and A C D, together eqnal to two ligUi angles. n For, draw C N at right angles to i ^ D E ; then D C N and N C E are two right angles ; that is, D C N, with N C A and ACE, are two right angles ; and since D C N and N C A nuike up D C A, therefore, UCA and ACE are two right angles. „ ^ k >ht lines, as A B, C D, inteisect or cross eaoli 116. If two straigiit lines, other, the o])i)osite and vertical angle; Since C E stands on A 13, the angles CEA and C E B are e(iiial c- to two right angles (No. ll;j). Again, since BE stands on CD, the angles C E B and BED are equal to two* right angles. Hence C E A and C E B are equal to BED and C E B. Take away the angle C E B, common to both these sums, and the remaining angles CEA, B E D are equal C E A, BED, are e<|Ual. (No. 11-2, 4). 117. If two triangles, as ABC, DEF, have two sides of the one, as A B, A C, equal to two sides of the other, as D E, D F, and have likewise the angles A, D, contained by those sides, equal, the two triangles are equal in all respects. For if the point A be laid on ^ D, and the line A B on D E, the point B will coiiici<le with E be- cause A B is equal to D E. Also, since the angles .A, D, are equal, the line A C will coin- cide with D F, and the point C of A C will coincide with the * point F of D F, because A C is equal to D F. 'I'hen since B coincides with E, and C with F, the base BC coincides with the base E F, and is therefore equal to it. Since therefore the three sides of the triangles are equal, the triangles are equal, and either laid on the other (two equal sides being laid on two e(pial sides) will exactly c >ver it. Hence the two remaining angles must be equal, or B is equal E, and C to F ; or, tiie triangles are equal in all respects. The above proves the mpthod No. 100. For .suppose A and I, B and I to be joined by linr.. then the two triangles C A I, GUI, h.ive the sides C A, A I equal to C B B 1, and the third Hiile rommoa Hence they arc e<iual, and the angles AC I, I C B l)eing eiiual, e»rh u Uir ol A C B. 34 INTjlODUCTlCM. 118. If two triangles ABC, DEF (fii,-. No. 117) have fu? angles B, C, in one, equal to two angles E, F, in the other respeciivcly, and also the sides B C, E F, adjacent to the equal angles, equal to oach other, the two triangles are equal. Suppose the point B to be laid on E, and the side BC on E F, the ])oints C and F will coincide because B C is equal to E F. Again, since the angles B and E are equal, the side B A will fall on E D ; and because the angles C and F are equal, the side C A will fall on F D. Hence, as the point A belongs to both the sides B A and CA, and D to ED and FD, the point A will coincide with D, and the angles A and D are equal. Hence the two triangles are equal. 119. In an isosceles triangle, as A B C. the angles B, C, opposite the 'jqual sides A B, A C, are equal. Suppose the angle BAG bisected by the ^ line A D. Then since A B and A C are equal, and the side A D common to the two triangles AD B, A D C, and the angle BAD equal to CAD, each being half of BAG, these two triangles are equal in all respects (No. 117), and therefore the angles B and C are equal. Cor. 1. Since the base B D is equal to the base C D, a line bisecting the angle contained by the two equal sides of an isosceles triangle likewise bisects tlie tliird side. Cor. 2. Also, since the adjacent angles A D B, A D C are equal, they are right angles, or the said line is perpendicular to the third side. Cor. 3. If tlie third side is equal to A B or B C, the angle A is equal to B or C ; or an equilateral triangle is equiangular. This proves tlie method No. 97 (1) ; for supposing C D, D I, and C E, E 1 joined, tlie two C D, D 1 are equal to C E, E I, and C I is common; hence the triangles are equal. Anc the angles D C I, E C I are equal, and each is half D C E ; hence C I bisects D C E and ii, perpendicular to A B. The like proof applies to No 97 (2) ; for suppose P 1, Q I to be joined ; then C P, C Q are equal to P 1, Q I, and P Q is common ; hence C P Q is equal to I P Q, a.id P 13 which Uius bisects C P I, is perpendicular to C D. The same kind of proof applies to Nos. 96 (3) and 9S. A, B, uil. IS IS 1'20. Every triangle which has two angles, coles: or tlie sides CA, CB are also equal. If C A is not equal to C B, let it be greater, and lalvo a part of A C, as A D, equal to C B. Then since D A, C B are equal, add to each of llicm AB, and the two DA, A B, are equal to liic two C B, A B (No. 112, 3). Also, since D A is equal to CB, the angles DAB, CBA are equal (No. 1 19). Hence the triangle A DB, having the two sides DA, AB, and the included angle DAB equal to the sides C B, A B, and the angle C B A, is equal to the triangle CBA (No. 1 17), or the less to the greater, whicli is absurd. Hence A C, C B are not unequal, that is, they are equal. OoR. If the third angle C is equal to A or B, the side A H miiet 3ft L'l-y L'nuiaiigiilar [IS A B, he in-oducct!, the citlier oi tlio iiiteruir and in like manner be equal lo C B, or to C A ; tliat i triangle is eiiuilateral. 121. If a side of a triangle ABC, ^'xlerior angle CBD is greater than .opposite angles A and C. Bisect C B in E, join A E and pro- duce the line till E ¥ is equal to A E ; and join ¥ B. Then since A E is ecpial to E F, and B E to E C, and also the angle A E C lo the angle B E F, the two triangles A E C, B E F have two sides and the included angle equal in each. Hence these two triangles are ecjual (No. 1 17), '• and therefore the angle C (o|)i)osite the side A E) is equal to th« angle E B F (o])posite the equal side E F). Hence C 13 D which contains C B F is greater than C. In like manner, by jnodiicing C B to a ]>oint G, and bisecting A B, it would be proved that the angle A B G, or its equal CBD, i.s greater than A. 122. Any two angles of a triangle are together less than iwo right angles. Produce the side B C of the triangle ABC,toD. Then the exterior angle AC U of the triangle is greater than the interior and opposite angle ABC (No. 121). Add to each angle AC B, then A C D and ACB, are greater than ACB and ABC (No. 112, 5) ; and since A C D, ACB are equal to two right angles, A C B, and ABC are less than two right angles. The same may be i)roved of the othei angles by producing the other sides. 12:$. If a straight line ABmeeting two other lines CD, EF, makes the alternate angles CG H, G H F ecpial, the two lines are parallel. 717- For if they are not, they will meet on one side of A B; let them meet at I, then G II 1 is a triangle, and the exterior angle (' C« H is greater than the interior and oi)i>osite angle (J II F (No. 121 ). But these angles are ecjual by the supposition, therefore the lines do not meet towards I. In like manner it may be shewn that they do not meet on the 36 INTKODUCTIOX. other side of A B, and heiicfi that they do not meet at all ; that is, they are parallel. It appears by fig. 2, that the lines meet on that side on which the two interior angles are less than two right angles. For I G H, I H G are together less than two right angles (No. I'i2). 124. If a straight line AB (fig. 1, No. 123) falling on two lines C D, EF, make the exterior angle A G D equal to the interior and o])posite angle G H F (on the same side of A B), the two lines are parallel. Also, if the two interior angles D G H, G H F, are equal to two right angles, the lines are parallel. The angle A G D is by supposition equal to G H F, and A G D IS equal to CGH (by No. 116); hence CGH and GHF are equal, and they are alternate angles, and C D, E F are parallel. Again, since D G H, G H F are equal to two rigiit angles by the supposition, and since CGH, D G H are equal to two right angles by No. 115, CGH, DGH, are equal to D G H, GHF; take away the common angle DGH, and the remaining angle C G H is equal to G H F, and they are alternate angles, therefore CD, f i F are parallel. 125. If a straight line A B (fig. 1, No. 123) fall on two parallel lines C D, E F, it makes (1.) The alternate angles C G H, G H F, equal ; (2.) The exterior angle A G D equal to the inteiior and opposite angle GHF; (3.) The two iulerior angles DG H, GH F, equal to two right angles. (1.) If CG H be not equal to GHF, let it be greater; add to each the angle DGH; then the angles C G H, DGH are greater than the angles D G H, G H F, and C G H, D G H are equal to two right angles (No. 115) ; therefore DGH, GHF are less than two right angles. But, by fig. 2, No. 123, this is the case in which the two lines meet at I, whereas they are here parallel by the supposi- tion ; therefore C G H is not greater than G H F. In like manner it might be shewn that it is not less ; it is therefore equal to G H F. (2.) Since AGD is equal to C G H (No. 116), and CGH to G H F, therefore A G D is equal to G H F. (3.) Hence, adding D G H to A G D, G H F, tlie two A G D, DGH are equal to the two D G H, G H F. But A G D, D G F are equal to two right angles ; therefore DGH, GHF are equal to two right angles. 126. Prop. The exterior angle, as A C D, of a triangle (formed by producing one of the sides of the triangle), is equal to the sum of the two interior and opposite a angles, A B C and B A C. Produce the side BC to D, and draw CE parallel to BA. Then the angle E C D is equal „ ^^ to A B C since B D meets the p;iriillei8 B A ami CE(No. 125). Again, tlie alternate angles BAC, A C K, formed l>y A C, wliicli crosses tlie same jiarallels, are i(iiia>' (Mo. 125). Hence ACE antl ECD are togetlier equal to BAC and ABC; tiiat is, A C D, which is made uirof ACE and E C D, is e jual to B A C and ABC. 127. Prop. The tliree interior angles of a triafigle are together pq(uil to two riglit angles (fig. No. 126). By the above projiosition, ACD is equal to the sum of ABC and B A C. Add to each A C B ; then ACD and A C B are equal to the three angles ABC, BAC, and A C B, (No. 112). But ACD and A C B are equal to two right angles, therefore the angles ABC, BAC, and A C B, are equal to two right angles. Cor. 1. In a triangle which has one right angle, the other two angles make up a right angle; each of tiieni, therefore, must be less than a right angle, and each is the complement of the other to 90°. Cor. 2. If two triangles have two angles in the one equal, re- H|)ectively, to two angles in the other, they will also have the third jr remainina; angles equal. 128. Prop. The greater side of any triangle, as A C, is opposite lo the greater angle A BC. C A being greater than A B, make A D * e(jual to A B, and join D B ; then since AD is equal to A B, the triangle ABD is isosceles, and the angles ADB and "/ '^ \ ABD are equal (No. 1 19). But A B D /<-.. \ which is contained within A BC is less than ^^ \ A B C. Hence A D B is less than A B C. '■ — J„ Now ADB is ecjual to the sum of A (J B and C B D (No. 125); hence ADB is greater than A C B, that is A B D is greater than ACB, therefore ABC is greater than ACB. In like manner, by taking C D equal to C B, it would be ])roved that the angle B is greater than the angle A ; and, by taking D on BC, and BD ecjual to B A, tiiat the angle A is greater than the angle C. 129. Prop. The line drawn [jerpendiciilarlv from a given point C, to a right line A B, as C D, is the shortest that can be drawn from C on A B. Take any point E in A B, and join C E. c Then since 'in the triangle C E D, C D E is ] n right angle, the angle C E D is less than a right angle (No. 127, Cor. 1), and there- fore (No. 128) C E is greater than C D. The same proof applies to any point whatever taken in A B. * '■ " ^ CoR As the angle C E D is acute, wherever E may be taken, thei'c is but one line which can be drawn periiendiculai- lo \ R from C. 'AH INTRODUCTION. 130. Def. a PaiiillLlograui is a four-sided figure of which the opposite sides are parallel. 131. The opposite sides ofa parallelogram, as A B, CD, are equal; also the opposite angles are eipial ; and the diameter, or diagonal. G B divides it into two equal parts. Since AB and CD are parallel, and CB meets them, tlie alternate angles ABC and BCD are equal (No. 125). Also, since A C, JJ D, are parallel, and B C meets them, the alternate angles A C B, C B D are equal. Hence the two triangles ABC, BCD having ^ two angles equal in each, and the side BC adjacent to tiiem com- iiioii, are equal (No. 1 18). Hence A B is equal to C D, and A C to B D ; also the third angle A to the third angle ofiposite, D. Since the two triangles are equal, and make up the whole figure, each is half the parallelogram, or C B bisects A D. 132. The straight lines C A, B D (fig. No. 131) which join the extremities of two equal and parallel lines A B, C D are theuiselves both equal and parallel. The triangles ACB, CB D, having the two sides AB, CD equal, and the side BC common, and also the included angles ABC, BCD ecituil, are equal ; hence AC and B D are equal. Again, since the other angles are equal, ACB and C B D are e(|ual, and hence A C, B D are parallel. Tliis (iroves the method No. 93 ; for the equal distances laid off from C and B |ipr[iendiiular to A B, form two sides of a parallelogram, of which the other sides al.so are parallel. And the like reasoning applies to No. 94. 133. Parallelograms, as ABCD, A BEF, on the same base A R and between the same parallels A B, C F, are equal to each otiu^r. Since CD and E Fare each equal r n e r to A B, they are equal to each other. V V .^^ y' Add to each D E, then C D, D E, \ Y / are equal to E F, D E (No. 112, 3), \ / \ / or C E is equal to D F. Also A C \ / \ / IS equal to B D, and A E to B F, \ / \ / nence A C, C E are equal to B D, \ /' \ / DF, and the angles ACE, BDF, ^ if are equal, because AC is parallel to B D (No 12.5). Hence the tri angle A C E is equal to the triangle BDF (No. 1 17). Take away the triangle ACE from the whole figure A B C F, and the remainder is A E F B ; again, take away the triangle B D F from the same figure, and the remainder is ABCD; therefore since t'.ese triangles are equal the remainders are equal (No. 1 12, 4), or the |)arallelograms ABCD, ABEF are equal. CoR. Parallelograms on equal bases, and between the sanii* parallels, are equal. For since the bases are equal, either of ti)em pliiced on the other will coincide with it, and the above proof npplies. GKO.METRY. ,W 134. A PaniUelograin A BC D is double of a trian<ile A H C od llic same base, A B, and between tbe same parallels, A B, C E. Draw A F parallel to B E, tlieu t k n ,, A B F E is a parallelograui, and it is e(|ual to A BCD (No. 133). Hence the triangle A BE, which is halfof'ABEF, is equal to half A B C D, or the paral- li;lograni is double of the triangle. Coil. Triangles on the same or equal bases, and between the same parallels lire equal. For parallelograms under these two conditions are, by No. 133, and Cor, equal, and the 'riangles being the halves of equal parallelograms, are equal. 135. Def. a Square is a four-sided figure of which all the sides are equal, and all the angles right angles. 130. Phod. To describe a square, A E, on a given line, A B. Draw A C perpendicular to A B, take AD ^ equal to AB, and through D draw DE parallel to A B ; and through B draw B E parallel to A D (or take D E equal to A B, and join B E). 'I'hen A D E B is a parallelogram, of wliich the opposite sides, being equal, are eacii equal to A B. Also since D E is parallel to A B, and AD meets them, the angles EDA, DAB, are equal to two right angles, and since A is a right angle, D is a riglit angle, and the ojiposite angles to these being equal to them are also right angles. 137. In any right-angled triangle, as A B C, the sq\iare B E, on the hypothenuse B C, is equal to the sum of tiie s(pia)LS Ci B and C I on the other two sides. Draw A K L perpendicular to B C, • or parallel to B D, which is perpendicular to BC, and join FC and AD. Then, since B D is equal to B C, and F B to B A (No. 135), the two sides F B, B C are equal to the two A B, B D (No. 112, 3). Also, the angles ABD and F BC are equal, since each contains a right angle and tlie common angle A BC Hence the triangles ABD and FBC are equal (No. 117). Now the triangle ABD is half the parallelogram B L, because they are on ne base B D, and between the til vNo. 134). Likewise the dince (1 C and 1' B are ])ai the square BC me e(|ual. Ill like luaiiiier, l)y juii same parallels BD, .A L riangle FBC is half the square B G, llel. Jlence the parallelogram B L and ug the jjoints B H, and .\, E, it would 40 INTRODUCTION. be proved that the parallel equal C L and the square C I are Hence the sum of the squares BG, C I, is equal to the sum of the parallelograms B L, C L, that is, to the square B E. Hence in a right-angled triangle if we have two sides we can always find the third: thus, suppose the hyp. is 100, and the base 64, the squares of these are 10000, and 4096; the ditf. of tlieso squares, or 5904, is therefore the square of the unknown side, which Is 76-8. The theorem above proves tliat the triangle of the dimensions in No. 96 (1) is right-angled Fnr 3, and 4, squared, are 9 and 16, and the sum of these, or 25, is the square ot 5, the third side. 138. The perpendicular on the extremity of the radius of a circle, as A T, is a tangent to the circle. Take any point D in AT, and join a „ ^ C D ; then since CAD is a right angle, CD A is less than a right angle (No. 127), and therefore C D is greater than C A (No. 128) or falls beyond the circum- ference, that is, AT touches the circle at A only. CoR. As only one line can be perpen- dicular to AT (No. 129), the centre of the circle must be in the line feri>endicular to the tangent. 139. The angle at the centre of a circle, as A C B, is double the angle at the circumference, as A D B, both angles standing on the same arc A B. .Toin D on the circumference and C the centre, and ])roduce the line DC to E; then the e.xterior angle ACE of the triangle ACD is equal to the sum of the two interior and opposite antjles CAD, and C D A (No. 126). But C A D is equal to C D A, because C A and CD being equal, A C D is an isosceles triangle (No. 119). Hence A C E at the centre is equal to twice A D E at the circum- ference. Again, the exterior angle B C E of the triangle B C D is equal to the sum of C B D and C D B. But these angles also are equal, because C B and C D being equal, C B D is an isosceles triangle ; hence B C E at the centre is equal to twice B D E at the circum- ference. r.F.OMETRr. 41 ^l0^v, ill fig. 1 (where the iliainetci- of tlie eirelo jjasscs clear o( die arc A B), ACB is tlie ditterence of B C E aiui A C E, and is duuble of A D B, the ditterence of B D E and A ]) E. When E falls on AB, as in fig. 2, ACB is the sum of ACE and B C E, and is double the sum of the angles A D E and B D E. or the an-le A U B. 140. The aniile at tlie circumference is measured by luilf the arc Siibtendinii' it (fig. No. 1:39). As A C B at the centre is measured by the arc A B, it is evident that ADB at the circumference (which, bv the prop, is half ACB) is measured by half A B. Thus, if A B is 58°, the angle ADB will be 29°, for any point of the circumference at which D may fall, except between A and B. Tliis proves th? method No. 100, for, since C A, AI (supposing A, I, and B, I, joined) t-eequal to CB, BI, and C I common, tlie triangles CAl, C B I are ec|ual, — hence ACI •Dd I C B are eijual ; eacli therefore is lialf of A C B, and is measured by half the arc A B. 141. The angle in a semicircle is a right angle. If the arc A B increases to a semicircle, A moving to E and B to D, A C and C B (fig. 1, prop. 139) falling into the same line, form H diameter, the angle A C B becomes two rigiit angles or 180°, and then A D B, or half ACB, is 90". Hence tiie angle in a semicircle is a right angle. This theorem proves the method No. 96 (2), for since 1 K is a diameter, the angle at M, t point on the circumference, is the angle in a semicircle. 142. The angle in a segment greater than a semicircle is less than a right angle. The segment B A C of the circle being greater than a semicircle, the other segment BDCmnst be less than a semicircle ; and the Hngle B A C in the greater segment being measured by half the arc B D (', that is, bv a quantity less than half 180° {No. 140), is less than a right angle. 143. The angle in a segment less than a semicircle is greater than a right angle. The segment BAG being less than a semi- circle, the segment B D C must be greater than a semicircle, and therefore the angle B A C, which is measured by half BUC (No. 140) is greatei- than Im.'f two right angles or than one right angle. 144. A line, CD, drawn from xuj chord A B, is jierpeiidicular to a circle bisecting 42 INTRODUCTION. Join C A, C B, tlien C A and C B are equal by the ilef. of a circle (No. 74). Also A D and D B are e((ual, each being half of A B, and C U is common to the two triangles CAD, C B D. These triangles, therefore, having their three sides equal, are equal ; hence the equal angles C D A, C D B, opposite the equal sides C A, C B, being adjacent angles, are right angles. » Con. The line from the centre bisecting the chord bisects the arc A B. For since the two triangles, as above, are equal, the angles A CD and BCD, opposite the equal sides AD, DB, are equal, and biiiig at the centre are measured by the arcs on which they stand. The above proposition is the principle of the method of finding the centre of a ciiile. 145. Triangles bavins the same altitude are proportional to their bases. The altitude is the peipendicular distance of the vertex, or summit, from the base. Let the base B C of the triangle A B C be divided into any niim- her of equal parts, as three, B^, g h, AC, and EF the base of the triangle D E F, into four like parts, E?, ih, kl, l¥, then B C is to £ F as 3 to 4. Join the points A//, A //, and Di, D ft, D /. Then the triangles A B^, Agh, AhC, and D E i, Bik, JMil, THY are all equal, being on equal bases, and having the same altitude (No. 134, Cor.) Hence the triangle ABC contains three parts, of which D E F contains four, and, therefore A B C : D E F : : 3 : 4, which is the ratio of the bases.* 146. A line DE parallel to a side B C of a triangle ABC divides the sides A B, AC, intiie same proportion, that is, AD ; A B : : A E : A C. • If it be impossible to find a quantity, or measure, B^, wliicli ahall divide B C and E F into an exact number of equal parts, as 3 and 4 above (that is, when B C ami E F are said to be incommensurable) we must take a smaller ([uantity, and a greater number of triangles j and by taking this measure sufficiently small we may make the error of using it ins^■ad of the true proportion as small as -re please. Join B E, C D Tlien tho ti inn-l.^s B D E. CI) Eon the same base 1) E, anil between tlie same parallels 1) E, B C, are equal (No. KU, Cor.) Add to each tiie triangle A I) E, then the whole triangle A B E is equal to the trianole A P C (No.'l 14, 3). Hence the triangle A B E : A B C : : A D C : A B C. Now triangle ABE: triangle ABC:: base A E : base AC, since they have the same altitude, viz the jierjien. lieular drawn from B on A C or A C ju-oduced (No. 145). Also, triangle ADC: Diangle ABC:: base A D : base A B, And tiie triangle A B E is equal to the triangle ADC, hence the two |>ro])ortions are the same, and A E : A C : : A D : A B. In like manner, as the triangles A D E, E D B, have the same altitude, viz. the perpendicular drawn from E on A B, we have triangle A D E : triangle E D B : : A D : D B. Also since the triangles A D E, EDC have the same altitude, viz. the perpendicular from D on A C, triangle A D E : triangle E D C : : A E : E C. But the triani?^les EDC and E D B are equal, hence A D : D B.: : A E : E C. This pi,)of applies to the sector. The line of lines on each leg is the side of an isoscele* lriaii){le, and the transverse distances 1,1, 2,2, &c., are the bases of so many isosceles tri- «nj;)rs ; the angles at these bases being equal, the bases are parallel, and the sides of the leveral triangles so formed are proportional. 147. Def. Similar triangles are such as have the sides about the equal angles proportional. 148. Equiangular triangles, as ABC, DEF, have the cor- responding sides about the equal angles jiroportional, that ia. A B : A C : : D E : D F. Jjet tlie angles A and D be equal, as also B ^^ and E, C and F. * n. Place the triangle DEF on ABC, D |\ \ being placed on A, and D E on A B, and let I \ I \ CJ be the i)oint where E falls. '■/ \" e — ' » Then since the angles A and D are equal, I \ and D E is on A B, D F will fall on A C ; let, \ therefore, H be the point where F falls. Then ^ \ since A (} H is equal to E, and B to E, A (i H is eipial to B, and the lines G H and B C, which make equal angles with A B, an' therefore parallel. Hence, by No. 146, A B : A C ; : D E : D F. Cor. Hence equiangular triangles arc similar (No. 147.) 149. In a right-angled triangle ABC, a line B D drawn from the riglit angle perpendicular to the l)y])otheiiuse, divides the triangle into two similar triangles A B D, B D C. t* INTRODUCTION. The triangles ABC, A D B, Iiaving " each a riglit angle, and the angle A common, /''^l' have the third angle also equal (No. 127), / ] \ they are, tiieretbre, equiangular. ^^^ \ For the like reasons ABC and B D C y^ 1 \ are equiangular ; therefore the two tri- ° ^ angles ABi), BDC, are equiangular, and tiia sides about the equal ingles are proportional (No. 148). Hence (1) AC: AB :: AB : AD. (2) A C • C B : : C B : C U. (3) AB: AD ::BC : BD.* 2. Terms of Trigonometry, 150. These terms occur in all calculations in \vh ftiig'les are concerned. 151. PN(y is a riglit-angled triangle; a quadrant is describe<l with the radius C P, from the^ centre C; C N and C P are produced, a:i(l A T is drawn parallel to P N. 152. The perpendicular P N, drawn from the extremity of the ai-c \ P, upon the radius C A, is called the sine of the angle PC A (to which it is op])ositc). When tl.e arc is very small, or P very near A, P N and A P, or the arc and sine, nearly nrincide. When the arc is 0, the sine is 0. When the arc is 90°, P falls at B, or the sine jf 90° is equal to the radius. Thus the sine is always less than tlie radius, though near 90" t becomes very nearly equal to it. 153. The line C N, between the centre and the foot of the sine, is called the cosine of P C A (to which it is adjacent). It is called cosine because its equal P n, is the sine of P C n, the complement of PCN. When the arc is small, N falls near A, and C N falls nearly on C A, or the cosine of a small arc is nearly equal to the radius ; for the arc 0, they are equal. When the arc is near 90°, the cosine is very small ; and the cosine of 90° is 6. Thus the than the radius, though it may approach indefinitely near to it. alnavs les * By (1) A C X A D = A B X A B, or, as it is written, A B= n\. A V C D = r B- ; henee the products A C x A D an.l A A K s,|uare and B (' square. )5ut A C x A D and A C x C 1). V D, or as A C x A V, which is called A C square; hence A C and B C sipuire. Tlie term square here denotes tlie number n l)y itself ; thus, if A B is ?.. A B- is 9. and this is the numbei Hie squore desi rilicd iiu A B. Uenoe this is another form of the propo; read A B square ; and I I) aiv In-.-tlier equal . saiiir as A { ' x A D al re is equal to A B squa s (ill the line) multiplii iqnare units contained No. 137. TRJ(.ONOAir.TUT. 46 15■^ Tlip line A T, drawn fVdni tlie extrfniity of one radius (as ». A), tdiic-liiiig the firclo, and meeting tli(M>tii(r radius produced, is failed the tangeyit of the angle P C A, or arc PA. N\'hen the arc is small, A T but little excoedn P N ; when the arc is the tangent is j wheu the arc is small, the tangent and sine may be taken for each other, and for the arc. When the arc is 90°, the tangent is intinitely great. Tlie tangent is less than the radius, »tcording as the angle is less or greater than 45°. The coUiiifjeiit is the tangent of P C n, which is the complement of PCN, and would be drawn from the extremity of the radius CB, meeting C P produced. l.OJ. The line CT meeting the tangent, is called tlie secnnt. Tiie cosecant is tlie secant of P C n, and meets the cotangent. When the aic is 0, the secant is equal to the radius. When the arc is 90", the secant in infinitely great. The secant is always greater than the radius, as is also the cosecant. 156. The line A I^ is called the versed sine. 157. These quantities are calculated for a radius of the same constant length, and to each minute or smaller division of the quad- rant, and are inserted in Tables. Then, since the sides of all right- angled triangles having the same angles are oronortional (No. 148), the tables afford the means of finding the relations among the parts of a right-angled triangle, of any kind or dimensions, by simple proportion. For example, the sine of ;S0** is ^ the rad. (see No. 159, Cor.), or 0-5, the log. of which, by No. 58 (2), is y-698970, as inserted :n Table 68. These are the ]irinciples on which the Traverse Tables and »1'» Trigonometrical Tables are consti-ucted. 3. Prop"siliom of Trifjomiwrtn I5S. The sine of an arc is half the chord of twi^ Take the area A P, A Q e(|ual to each other, and join P Q. Then the angks PC A, ACQ are equal (No. 82). And since CP=CQ, and C M is common to tlie two triangles C P M, CQM, these trianirles are equal (No. 1 17); hence PM = ]\rQ; therefore P M, the sine of A P, is half PQ, the chord of twice A P. 159. Tlie chord of 60"' is equal to the radius. Let A P and AQ (fig. No. 158) be each :iO<', then the is m"; and since tlie three angles of the triangle PCQ ar< 180° (No. 127). C P Q and C Q P are together equal to 120° ' Also, since CP = ('Q, these two angles are equal (No. 119), and each, therefore, is CO*. The triangle is, therefore, equiangular, and eon- wquentlv, e<iui'ateral. No. 120. Hence PQ = C P. Von' Since P M is half P Q, it is cpiai to half C P : or the sine ol 30°, which is the cosine of 60°, i-^ half liie radius. 4r> INTROUUCTIOV. I GO. TI)(! secant of 60° is equal to twice the radius. Since I' N and AT are both perpendicular to G A, tlioy are parallel (No. 124), and the triangles C P N, CT A, are similar (No. 148), hence CT:CP::CA:CN, that is, as rad. : cos. 60*>, or as 1 to ^ that is, as 2 : 1. 161. The tangent of 4.j** is equal to the radius. Let P C A (fig. No. 160) be 4.5°, then C T A is also 45** (No. 127), hence the triangle is isosceles and tlie sides C A, AT are equal. ' ;" Con. Hence also, by similar triangles, C N = N P, or the sine and ens. of 45° are equal; as are also the tangent and cotangent. zh 4. Construct/ ti<) the Ciinoiis, and working them by Logarithms. 162. Take a right-angled triangle, as A B C, and supjiose auothtr similar to it, as PNC, drawn in a quadrant, as in No 151 ; then CA: AB ::CP: PN; that is, C A -. A B : : rad. : sin C (by 152). Tiie second triangle, PNC, is, in fact, here referred to for illustration only ; for it is evident, without it, that C A and A B themselves stand in tlie same relation to each other as that of radius and sine; hence By No. 1.52. CA : AB By No. 153. CA : CB By No. 154. CB : BA By No. 155. C B : C A 16.3. It is easy to recollect these analogies, each of which begins with two sides, by observing these condition-s. 1. One of the three sides must be made radius, and the analogy always begins with that side. 2. The other sides will then become sine, cosine, tangent, cotan- gent, secant, or cosecant, of one or the other of the two acute angles. The figures below sufhcieiitly illustrate the application of tho terms. c N rad. : sin. C. (1.) rad. : cos. C. (2.) rad. : tan. C. (3.) rad. : sec. C. (4.)* * The learner will much more speedily apprehend the purposes which the expressions of trigonometry answer in the sciences of calculation by considering these proportions as reprc- •enling the change of quantities in a certain ratio, as in No. 48 (.S). Thus AB in C \ dimiuKhed in the ratio of the sine of C to 1 ; C B in that of cosine to 1. A B is alKi (' B diminished or incrmsnl in thr ratio of tan. to 1, according as C is less or greater than 44° tnd C A is C B iticreuscd m the ratio of secant to 1. TRIOONOMETRT. 4/ To employ riglitly the tcnus .sine, cosine, kc, observe — 3. Tliat wlifin the hypothenune, or longest side (whicli is ojjposile llift riji'lit angle), is made the radius, The side opposite either of the aciife ati<>Ips is the sine of that angle; and the side adjacent to eitiicr angle is the cosine of that angle. 4. Wlien either of the sides eontaining the right angle (or leijs* as they are called), is made radius, the other side becomes tlie Utntjent of the angle opposite to it; and tiie hypothenuse becomes tlie tecnnt of that angle wiiicli is coutnined or included between tVieZ/'and tiie radius. The learner should be able to construct the above analogies (which he ^viIl find very easy; before he proceeds to the solution of any question, without regard to what is given or what is not given. 164. We now proceed to the calculation of a problem. 'J'lie above analogies or proportions consist of four terms each. Hence, if three are given, the fourth may be found (No. 46). But the radius is assumed in the trigonometrical fables as 1 (which is the simplest of numbers), and iieiice, of the three remaining terms, if two are given, the third may be found. Hence, in any right-angled triangle, consisting of three sides and two angles besides tlie right angle, if two parts which enter into any one of the above analogies are given, the third term of that analogy may be found. 16f>. The proportions may be solved by multiplication and divi- sion ; thus, suppose, CA (hg. No. 162) measures 37 feet, and the angle C is 29° .52', and we want to find A B. We have by No. 1G2 (1), C A : A B : : 1 : sin. C, whence (No. 46) A B = C A x sin. C (the 1 not being written). Now the sine of 29° .02', given in tables of natural sines (of whicii the loqs. are given in Table 68) is 0-498 nearly, hence A B=37 x 0-49,S = 18'426. But in order to save such tedious processes, logarithms are em- ployed in the manner described, No'^. 64 and 65. Thus, A B=37 X sin. C, becomes log. of A B = log. of 37 + log. sin. C. Again, if CA were re(|uired, aiul A B given, we should have CA = ABx l-=-sin. C; or, (supprcssiim- the 1). log. C A = log. AB-log. sin. C. 'l"hc following rules are deduced from these principles. The learner will do «ell to verify all his work by the Traverse Tables. Tliis proceeding is described in the explanation to the Traverse Tables. 166. The rule for working any analogy by logarithms is very simple, and there are but two cases: 1. In which it is re<|uiied to find one of the mean terms; and, 2. In which it is requireil to fin<l one of the e.xlreme terms. : also railed the ban bikI firrjiendieiilar ('S'.\ 89). Tliese terms, bfiun lies wtiicli lire liori/diitBl an. I vertical, lu the reader holds thn liguni ved ciilinly at coiiveniencc. 4K INTRODITT'ON. (1.) To find a mean torni. Atld together tlie logarithms of the two extremes, ami subtract from tlie sum tlie logarithm of the other rueiin. The rt:mainder is the logarithm of the term required. (2.) To find an extreme term. Add together the logarithms of tiie two means, and subtract from tlie sum the logarithm of the otlu'i extreme. The remainder is the logarithm of the term required.* JYote. — The log. of the radius (as employed in the analogies) is 10, this being used for convenience, as stated at p. 19, note +. Case T. Given the angles and the hypothenuse, to find the two .sides. Ex. B is the right angle. Tlie angle A is 50° (whence C is 40°, because the two acute «iigles are together 90°. (See No. 127, Cor.) C A is iS feet. It is required to find B C u-id B A. V,'e must employ two sides, and one of them must be the unknown, or required side • If C A, llie hypothenuse, be radius, C B b«roiiifs the sine of A (No. 163), hence C A : C B : : rad. : sin. A ; in which C B, a mean term, is required. »™ce, by No. 166 (1), we have to add the logs, of C A and sin. A, and subtract the CA 28 log. (tab. 64) I "4472 log. sin. (tab. 68) 9-884} log. 11-3315 CB = 2i-4 log. . ^3iS AB=iS-o ght have used C B a s cos. C, We might have used A B C A : C B : : rad. : cos. C, that is, C A : A B : : rad eCB : CA :: rad. : sec. C. otherwise A B : A C : : rad ^■ If C A, the hvpothenuse, be radius, becomes the cosiiie of A (No. 163). C A ; A B : : rad. ; cos. A ; in which A B, a mean term, is requ Hence, by No. 166 (1), we have to ad: !og8. of C A and cos. A, and subtrac log. (tab. 64) I- log. cos. (tab. 68) 9- log. II- CA: * It is necessary to remark here that the process .above differs from that followed i J iaamen in general, the object of which is simply that the required quantity may stand last. The example in Case III. by that method stands thus To find the Anglt As the hypoth A B Is to radius So is the perp. A C ^-7" 33 Tofim the side B C. As rad. lo-oooo Is to hypoth. So is sm. A AB 2-3430 9-94-!' ii-igoS ToBCi95-4 2-2908 Now the method proposed is more natural than this last ; because, when the two sides are taken together, their trigonometrical relation to each other is immediately perceived, which, -when they are separated, is not so apparent. Again, since the term sine, or cosine, is determined altogether hv that side which we make radius, the;-c terms should, arrnrdine >■< (he natural prn,;ress of ideas, immeilMelf/ folirm Ihc term riirliu... The mrthivt l'ol!o.rcd •! ilio shorter aud more elegant. Moreover, the nietliod just <;uoted, not being employed in TRIGONOMKTIIT. Case fl. Given the angles and one leg, to find the hypoilicnuM d the ullier Ic^. C is 171. Find A B and A 0. To find A C. Ti.lfe two sides, AC, C B, m«it« AC dius ; Ihcn, by No. 103 (3). Ex. C is 90°- Angle A is 30° 14', hence B is y)° 46', To find A B. T:ike the two sides, A B, B C make A •he hvpothenuse) radius ; then, No. 163. -^ 1 C : : rad. : sin. A ; A C : C B in nhich .\ B, an extreme term, is required. in which AC, an extreme term, is required. Hence, by No. 166 (2), we have to add Hence, by No. 166 (2). •Jie logs, of B C and rad., and subtract the C B 171 log. log. of sin. A.* A 30° 14' log. tan. — 9 7655 IC171 log. + 10, 12-2530 AC = 293-4 loss- 1-4675 A 30° 14' log. sine —9^020 AB=.339'6 log. 2-5310 I This might, like Case I., be worked differently. Thus, to find A B, we may make B C radius ; then A B : B C : : rad. : cos. B. Again, to find A C ; making B C radius, we have A C : C B : : ra'l. : Un. B. We might also, having found one of the unknown quantities, employ this quantity as a means of finding tiie rest; but in general it is better, when practicable, to depend only on the original quanti- ties given. Case III. Given the hyjiotlienuse and one leg, to find the angles iind the other leg. Ki. Angle C is to°, B A = 220-3, AC-- 101-9 "• ''"'' ""^ '"'K'* B- »"'' "'^» B C. To find B. To find B C. T«> ing the two given sides, ne have Takini; Ihe two sides, B C, C A, wo h»T« B A : A C : : rail. : sin. B ; 1 w'.iir^h sin. B, in extreme term, is required. AC 101-5 log. + 10, 12-0082 tiXziOi log. — 1-34 -,0 B = 27^ 33' sin. 96652 li C : C A : : rad. : tan B ; in which B C, an extreme term, is required. CA 101-9 log. + 10, I20tS2 B 27' 33' log. tan. — 9-71-4 B C= 195-4 log. 2-291 ' (Here, in computing by the canons, we arc obliged to employ B, as found.) »ny other srieiitifln process, every seaman who may require to e.xfend his scientific knowledtrs of these subjects will have to unlearn it and to adopt the other. The rules laid down abo>« icill be found, after very little practice, simpler and more intelligible, and therefore easier to recollect, than those of the old method. • Instead of subtracting the log. sine, cosine, and tangent, it is the same thing to aild ihe lof cosiT-nnt, snrarit, and cntanffent, because those la^it .ire the arithmetical comjdcmentt n the fir^i. We have oniitt«d tnis Ui the eiupiiplcg, to avoid confui-ing tlie learner. 50 INTttODUCTION. Case IV. Given the two legs, to find the hypothenuse and the UMules. Ex. The angle C (fig. in Case III., only marking B C as given instead of B X) is 90'- B C= 195-4, C A=ioi-s: fiiiJ B A and the angle A. To find angle A. AC : BC :: rad. : tan. A. Hence, by No. 166 (2), BC J95'4 log. + 10, I2"2909 AC 101-9 log.— 2-0082 A = 62°27' log. tan. 10-2827 Mnd B=.27 33 To find B A. Making B C radius, B A will become f>ta secant of B ; hence, BC : BA :: rad. : sec. B. Hence, by No. 166 (1), BC195-+ log. 2-290? B 27° 33' log. sec. io-c;23 BA = 220-3 log. 2-3431 As 10 is to be subtracted it is omitted in the index 11. A 39° 22', wlience C is 50° 38', required A 8 Ans. AB is 1II-3, andBCgi }. Ei. 4. Given the base AB 208, and angle A 35° 16'; find the hypoth. A C and the perpend. BC. Ans. AC = 254-S, BC=i4--i. El. b. Given the hypoth. A C 272, and base A B 232. to find the angles A and C, and B G Ana. A = 3i°28', C = 58'^ 32', BC = i4i. Ex. 6. Given the hypoth. C A 980, and base B C 720, required the angles and remaining leg. Ans. A 47° 17', C 42° 43', A B 664 «. VI. Mefhods of Solution.* 167. The solution of a qnestion in wlncli tlie result is re(iiiircd in nnmbers is obtained in three ways, namely, 1. Inspection; 2. Cal- cidation or Computation ; 3. Construction. (1.) Inspection usually implies taking out, ready calculated, from R table, the result corresponding to the elements of the particular question proposed. Tiie term has, however, a more general accepta- tion, being applied to the taking out, not merely of tlie result itself, but of quantities which compose it. This method being easy and expeditions, is the best for general practice when precision is not required; but as the tables adapted to tiiis kind of solution are necessarily limited, it is, on many occasions, iiot sufficient. ("2.) The general term Computation may be applied to every mode of solution by the composition of numbers only. Since, how- ever. Inspection includes the simplest cases of this kind, namely, iliuse in which either the required quantity itself, or the parts coni- * The matter in this section is, from its nature, adiijitrd .)iily tn the reader whi has made louie [irugiess in the subject. IMKODUCTION. 5UA The Solutiok of Obltque-ajtoled Plank TRiANOLEa. Case I. lu any oblique-angled plane triangle, given two sides and an angle opposite to one of them, or two angles and a side opposite to one of them, the remaining angles and sides are found by the following simple proportions : — As one of the given !>ide3 : sin. of its opposite angla :; the other given bide : sin. of its opposite angle. To find an angle, brgin with a side opposite to a kc angle Again, as sin. of one of the given angles : its opposite aids :: sin. of the other given angle : its opposite side. To find n side, begin with an angle opposite to a known side Ex. I. In the triangle A B G. given A C B 41" 13', iC 2S2 yards, and A B 2loyards, to find the rest. Now AB 210 being less tlian AC 282, the case ia mbigiious, and there :ire two solutions. At point C in the line BC make angle BCA = 4I°I3', 1 C l.iy off C A = 2S2, Hnd from A lay off A B = 210, C culling B'C in B and D, join AD. To find A B C and ADC. Ab AB 210 log. 7677781* ;ACB4i°i3' log. sin. 9 818825 ::AC 282 log. 2450249 : ABC 62° 14' log. sin. 9-946855 A BO = A DB.-. AI)B = 62° i4'-iSo = ADC 117' 46' ADC = ii7°46' + ACB = 4i=' 13' = 158° 59' - iSo" = DACai" i' ACli = 41° 13' + ABO = 62° 14' = 103° 27' - iSo° = BAG 76=33' To fin. I I! C. As AC B 4I°I3' log. cosec. o t8ii75» : A 1! 210 log. 2-322219 :: BAC76°33' log, sin. 9 987922 log. 2491316 !C 310' To find D C. As A C B 41° 13' log. cosec. 0-181 175*' : A B 210 log. 2 322219 :; DAC 21° i' log. sin. 9-554658 : DC 114' log. 2058053 ♦ Sf« note to p. 49 on the " Arithmetical Comiltment." 5lA PLANE TlilGOXOMETUr. Case [I. In any oblique-angled plane triangle, given two siiiea and the included angle, to find the rest. As the sum of the given sides : their difference :: taa. i sum of the unknown angles : tan. i their difference. The f difference being added to J sum will give the gre:iter angle, and bein' subtracted from it will give the less. The greater angles will be opposite the greater side. Es. 2. In the triangle AB C, given = 512 yards, (,- = 907 yards, and B 49' 10', to <-907+« 5i2 = c + (! 1419, c 907-11 5i2 = f-(t 395, }'. 49° lo'-iSo = A + C 130' 50' -=- 2 = '^ 65^ 25'. To find A and C. ia + c 1419 log. 6S4SolS» «-c 395 log. 2-596597 — ^ 65°2S' log. tan. 10-339642 ■3I°I9' log. tan. 9 784257 AsA 34° 6' log. cosee. 0251345* : a 512 log. 2-709270, :: B 49°io' log. sin. 9-878875 . A 691 log. 2-839490 31° 19' = C 96° 44', and 65" 25' -31° 19'^ A 34' 6'. Case III. In any oblique-angled plane triangle, given the three sides, to find the angles. From the half sum of the three given sides (S) subtract the two sides containing a required angle. To the logs, of these numbers add the arithmetical complement of the logs, of the sides; the sum of these 4 logs., rejecting 10 from the index, will be the log. sin. square, Table 69, of the required angle. Ex. 3. Inthe triangle ABC, given rt = 6, ft = 5. «nd<T = 4, to find the rr-«t. a + 6Hc = 6 + s + 4=i5-=-2 = 7-5(S), H Ji-b s^ 2^. S7S-4-3-: To find A. .'■--A 2-5 log. 0-397940 R-r3-5 log. 0-54406S f> 5 Ar. Co. log. 9-301030 c 4 Ar. Co. log. 9-397940 A 82°49' log. sin. square 9-640978 note to p. 19 on the " i tical Complenient," METHODS UK SOLUTION. 51 pDsi'iig it, arc tiikeii from tables, tlit- term Computation will lie i-iu- ployed in other cases, and always wlu-n precision is required and logaritluns are cnncerned.* (3.) Construction implies (in our present subject) drawing a figure of the actual case on a convenient scale, and in the proper proportions, the number of parts contained in tbe quantity required to be measured being taken from a scale adajited to the purpose. This process is tedious, and not, in general, capable of much pre- cjision, but it is the most readily intelligible of the three methods, and is, therefore, the least open to mistake. The seaman should, accord- ingly, be able to produce a figure of every case that admits one, and nhould acquire the habit of referring to the figure, in the mind as the only real security against mistakes in his work. The figure or natural representation of the case is, moreover, the foundation of the mathematical treatment of the question. 1 . Limits of Methods or Obacrvatioiis. 168. In every process of calculation, the elements which enter into it, and which are either observed at the time by instruments, or taken from tablas, are liable to eri-or. Every result, therefore, is, to Some extent, uncertain ; but the amount of error of the final result of this kind are usually divided into " rigorous" and " approximate," oi Indirect, as the latter are also called. In all solutions, however, we either deal directly with the quantities themselves, as arcs, angles, &c., in tlieir entire or integral state, or we corn- putt a difference from a certain value assumed or given, and thence find the required quan- tity. This last i)rocess is indirect, but tlie former may be effected indirectly also. Tha terms Integral and Differential would then, it is presumed, be more satisfactory, for the degree of appro.ximation obtained is altogether beside the question of the charai-ter of the solution. We do not, however, on the present occasion, depart from the usual terms. Wo •hall merely add, as some indistinctness prevails as to the properties of these different noiutions, that both are equally affected by errors of observation (as must of course follow, if they be both true), and thus the essential distinction between them, in practice, lies lu the different numbers of figures which they respectively require. There is another point on which we shall take the opportunity to make some remarks tnf the satisfaction of the scientific reader. In the present subjeci we are obliged, in most cases, to consider the required quantity, though really unknown, as if it were given, as it is an indispensable argument in reducing the elements ; — thus, in finding the longitude by clironii- meter, by the sun, we must assume a longitude in order to deduce the declination and eqna- t'on of time. Such solutions are, therefore, solutions by assumption, and the question naturally arises, What is the criterion by which to know whether the result is nearer tlie truth or further from it than the temporary value employed .' In general we have to solve, not the equation K=/(.r, y, z), but K, — ^ (x, y, z, u'), in whit'h «' is an assumed value of u , and w, a first approximation. The second approximation U iij=y (jr, y, 2, K|), and so on. Now, it is evident, without examining the successive difTerences u' — !/,, ", — Hj .... that the process is convergent, if u varies more slowly than u', that is, when -j-, < i. This is the case with all our problems within the limits i When 3-,>'. tl'e process is divergent, or the results are worse and worse ; and when ■= i, the assumption is reproduced. Again, when — / is positive, the results are all greater or all less than the tnith ; when negative, they are alternately Ino great, and too small, llenre. in (rneral. it depends on the data, and not on the greatness or smallness of the error o( niisnmption, whether the process ronverge oi not. The above, however, ajiplies, in strirt- 'ncii, only to tmull errors of assumption ) for largo cirora liighcr terms must be ciinsidrrcd. 52 INTUOOrCTION. oansed by an error in anyone ot'tlie data (or quantities given forilie solution oi' tho question) is very dift'erent under different circum- stances, being in some cases scarcely perceptible, while in others it may far exceed the very error to whicli it is due. If we agree beforehand that a probable error of observation shall not cauFC an error beyond a certain amount in the nisult, we must exclude all those cases in which it would produce a greater effect. and we thus assign li-aits to the method or observation. 169. Generally speaking, every element that enters into the computation is liable to error, and, therefore, each element will iuive its own indeiiendent influence in limiting the observation ; that is, in strictness, there will be different limits for each sejiarate element , but, for practical purposes, it is enough to assign the limits according to that element of which the error is most important. For instance, in finding the time by a single altitude of a celestial body, we employ its altitude and declination, and the latitude of the place. Now tiie latitude will often, and tlie declination sometimes, be correctly known, but the altitude can never, from various causes, be exem|)t from suspicion of inaccuracy ; besi<les, in general, an error of altitude ]M-oduces a greater effect on the result than an equal error in lati- tude or declination. Hence we limit the method of " time by an altitude off the ineridiair in respect of altitude only; and assuming that 1' error of altitude shall not cause more than 10' error in the time, we limit, for the more frequented latitudes, the celestial body to a ceitain bearing. 2. Degree of Dependutice. 170. The result of every computation is, as above remarked, No. IfiS, more or less uncertain. If we knew the error in one of the elements, we could easily find the effect it would produce on the result, by working the coni)Hitation over again; and if, under the circumstances, such error in the data is not likely to exceed a certain jpiantity, we should thus find the limit of probable error ;* for ex- ample, suppose in finding the time, the error of altitude is not likely to exceed 2', and that the efi'ect of this in working over again is 9', we say that 9' is the limit of probable error. 171. Since all the elements are more or less uncertain, there is a limit of probable error or degree of dependanee in respect of eacii. Hence the extreme probable error of the result is the sum of all these errors, supposing they lie on the same side. But, in practice, they will, in general, tend to neutralise each other, and it is enough to estimate the degree of dependanee in respect of the most important of them. 172. In some cases a small error of observation will produce a very great error in the result ; in others, a large error may not pro- * Tlie term " Degree of Dependanee " is preferred liere to " limit nf proljahle error." becBiise it describes in direct terms tlic appUcutioa or use iit' tliat limit, wliiili is, tu point ojl hew iicar tlie result may be depended upon. MFTHODS OF SOLUTION. 53 diice a sensible effect. For example, an error of 1' in the liiiinr distance, causes an error oi".i(Y or w in tlie longitude, wiiilean error of several miles of latitude may not, in certain cases, produce an error worth notice in the time as found by an observation. As i<o nicety in the mere working of the computation can, in any way, meet or counteract errors of observation, it is necessary, in forming a true judgment of tiie place of the ship, to try tiie effects of j)robablu errors; in other words, to try the degree of dependance. Thus, in the exam|)le of the lunar alluded to above, a novice might conclude that his longitude was, to the exact minute and second, that found by contputation ; but a more experienced computer, knowing that all liis elements are not absolutely correct, and that his result can scarcely be perfectly exact but by an accidental compensation of errors, makes an allowance for error; and assuming that the distance may be too much or too little by 30", for example, considers the observation as merely having established with certainty the ship's place within 15'E. or W. of the position deduced. 173. But the degree of dependance, besides being indispensable to rightly judging of tiie true place of the ship, or, ratiier, of the space on latitude and longitude within which she is to be found, has another important application, as it governs the amount of labour bestowed on the computations. For example, if the latitude is uncertain several miles, it is at once evident, that to proceed with as mtich caro and precision as if it were ascertained to a few seconds, is mere waste of time. Similar remarks have already been offered in the Preface, and they are particularly directed to the student's attention, who should l>e early impressed with the importance of improving his judgment by continual exercise, instead of trusting on all occa- sions to a mechanical routine of computation. 174. It is worth wliile to notice, that in working to a certain de- gree of accuracy, as, for example, to minutes, it is generally enougli to employ the neares* whole minute; but when one of the quantities varies very rapidly, it may be proper to work closer ; for it is easy to see that the inaccuracy of half a minute in a quantity which is multiplied by a number greater than 1, is increased, and apjiears 33 a whole minute. [l.j Personal Error. 17.">. The several errors to which each observation is exposed, and which accordingly enter into the estimation of the degree oi dependance, are described in their proper places; but there is one ^liich, though sensible only in cases where a considera])le step has been made towards precision, is of universal application, and i.s, Jierefore, ))ro])erly noticed here. It is found that ditferetit persons lio not agree in the precise instant of observing the same phenomenon. Again, some persona nre in the habit of observing more or less closely than others. The kind of error which is obviously present in such cases, is called the jerhonal ciror, or etjuiiticn. St INTRODUCTION. 'I'wo observers have been found to differ 0' 4 in tlie sun's transit over tlie wire of a telescope. 176. Wlien two images, in contact, lie stationary before two observers, it is difficult to understand why one of tliein should see them overlap, or the other open, or wliy they should not agree in the measure. But when the images are in motion, the observer's anxiety is roused lest he may miss the observation, and the excite- nient may lead him to think that he sees tlie contact before it really takes place. Hence there is reason to believe that the personal equation is, in some degree, a mattei' of tempei'ament. It also seems well ascertained that the personal equation is not the same for the same individual at all times, and that it is greatly influenced by fatigue, by the effort of observing, and, in fact, by every cause that affects the nervous system. It may, therefore, be advantageous to bear these circumstances in mind preparatory to undertaking observations in which much accuracy is required. 177. The existence of this error shews that when much precision is required, observations taken by different persons should not be mixed together until cleared of ])ersonal errors, since they may at the outset be ])resumed to be affected by unaiual errors ; and it is probable that many discrepancies are due to this cause, in cliHerva- lions whotUer by tlifl same or different observers. 56a Spherics, Definitions and Principles. Sphrrics is that part of mathematics which treats of the positiong and magnitudes of arcs of circles described on the surface of a sphere. A Sphere is a solid formed by the revolution of a semicircle about its diameter; this diameter is immovable during the motion of the semicircle. The Centre and Axis of a sphere are the same as the centre and diameter of the generating semicircle, and as a circle has an indefinite number of diameters, so a sphere may be considered to have an indefinite number of axes, round any one of which it may be conceived to be generated. Every Section ok a Sphere made by a plane passing through its circumference is a circle. A Great Circle is formed by a plane passing through the centre of the sphere. A Small Circle is formed by a plane that does not pass through the centre of the sphere. A sphere is therefore divided into two equal parts by the plane of every great circle, and into two unequal parts by the plane of every small circle. The I'ole.s of a Circle of a sphere are those points on the sur- face of the sphere which are equally distant from the circumference of that circle. Thus the poles of a circle are the extremities of that diameter or axis of the sphere which is perpendicular to the plane of that circle. All points in the circumference of a great circle are equally distant from both its poles. Small Circles of the sphere are those circles which are unequally distant from both their poles. The Poles of every great circle are each 90° distant from that great circle on the surface of the sphere, and no two great circles can have the same poles. The Diameter of every great circle passes through the centre of the sphere, but the diameters of small circles do not pass through the centre. Thus the centre of the sphere is the common centre of all its great circles. Parallel Circles of a sphere are those small circles the planes of which are parallel to the plane of some great circle. All parallel circles have the same poles, and may be conceived to be conc.entric to the gnat circle thf-y are parallel to. 56a introduction. A Spherical Angle is the inclination of two great circles of the sphere meeting one another. It is measured bv an arc of a great circle intercepted between the legs of that angle, 90° distant from the angular point. A Spherical Triangle is a figure formed on the surface of tlie sphere by the intersection of three great circles. The SHORTEyr Distance between two points on the surface of a sphere is an arc of the great circle passing through those points. The Stereographig Projection * of the sphere is such a repre- sentation of its circles upon the plane of some great circle, and thence called the plane of projection, as would appear to an eye placed in one of the poles of that circle, and thence viewing the circles of the sphere. The place of the eye is called the projecting point or lower pole, and the pole opposite is called the opposite or exterior pole ; also the projection of any point on the sphere is that point in the plane of projection through which the visual ray passes to the eye. The Primitive Circle is that great circle on the plane of wliieh the representation of all other circles is supposed to be drawn. A Right Circle is one which is perpendicular to the plane of the primitive circle, and, if it be a great circle, its plane passes! tnrough the eye and it is seen edgewise, consequently it is represented by a straight line drawn through the centre of the primitive circle. An Oblique Circle is that which has its plane oblique to the eye, and is represented by a curved line. Spherical Trigonometry is the art of computing the measures of the sides and angles of such triangles as are formed on the surface of a sphere, by the mutual intersection of three great circles described thereon. A Spherical Triangle has three sides and three angles. A Right-angled Spherical Triangle has one right angle. The sides about the right angle are called legs ; the side opposite the right angle is called the hypothenuse. A Quadrantal Spherical Triangle has one side equal to 90°. An Oblique Spherical Triangle has all its angles oblique. The Circular Parts of a triangle are those arcs which measure its sides and angles. Two spherical triangles are said to be supplemental to one another when the sides and angles of the one are supplemental of the sides and angles of the other, and one in regard to the other is called the supplemental triangle. Two arcs or angles when compared together are said to be alike when both are less or greater than 90°. But when one is greater and the other less than 90°, they are said to be tinlike. In every spherical triangle equal angles are opposite equal sides, und equal sides are opposite equal angles. Stereograpliie means representing a S'.lid on a plane eurlace. sniKKicAL trigonomethy. B7a Anj two sides of a spherical triangle are together greater than tit- third side. Each side of a spherical triangle is less than a semicircle or 180°. In every spherical triangle the greater side is opposite the greater angle. The sum of the three sides of a spherical triangle is less than 360°. The sum of the three angles of a spherical triangle is greater than two right angles and less than six, or always will fall between 180° and 540°. In right-angled spherical triangles, the oblique angles and their opposite sides are of like affection ; that is, if a leg is less or greater than 90°, its opposite angle is also less or greater than 90°. In right-angled spherical triangles the hypothenuse is less than 90° when the legs are of a like kind; but greater than 90° when the legs are of a different kind. In any si^herical triangle /\s sine of either angle : Bine of its opposite side :: sine of another angle : eine of its opposite sid*, Eight Spherics. The celebrated Lord Napier, inventor of logarithms, contrived a general rule, easy to be remembered, by which the solution of every case of right-angled spherical triangles is readily obtained. In any right-angled spherical triangle there are five parts beside the right angle — viz., two legs, two angles, and the hypothenuse. The two legs, the complements of the two angles, and the comple- ment of the hypothenuse are called circular parts. In any case relating to right-angled spherical triangles three of these circular parts arc concerned — viz., two given and one sought. If the three concerned are all joined together, ignoring the right angle, the central one is called the middle, and the other two adjacent parts. But if only two are joined together these are called the opposite, and the other the middle part. These being known, all the cases of right-angled spherical tri.'ingles may be solved by Napier's rules. 1. The product of radius and sine of the middle part = the pro- duct of the tangents of the adjacent parts. 2. The {iroduct of radius and sine of the middle part = the pro- duct of the cosines of the opposite parts. N.B. — As an aid to memory the letter a occurs in tangent and adjacent; and the letter o in cosine and opposite. In the following examples, insteatl of subtractinf/ the log. sine, cosine, and tangent, it is the same thing to add the log. cosec, sec, and cot.; becauso thc'^e last are the arithmetical complements of the first (see note, p. 49). 58 A INTRODUCTION. Ex. 1. Ill the riglit-ftrgled sphe 40" 30', B 90°, to find the otiier parts. triangle ABC, given C 61° 50', R C {a) To find A. Had. COS. A = Bin. C . cos. a CCS. A = sin. C . cos. a C = 61° 50' log. sin, 9-945261 = 40 30 log, COS. 9 88 1 046 A = 47 54 log. COS. 91126307 To find A C {/>). Rad. COS. C . = cot, b . tan. a cot. b — COS. C . cot. a « = 4O°30' C==6i 50 4 = 61 4 log. cot, 0068501 log, COS, 9:673971 log. cot. 9742478 E.>!. 2. In the right-angled spherical t; 13° 26', B 90°, to find the other parts. To find G. Ead. sin, c = sin. i. sin. C sip. C = sin. c . cosec. J A = 1 1 3° 26' log. cosec. 0037383 c= 50 40 log. .sin. 9888444 C= 57 28 log. sin. 9925827 To find B C («). log. spc. 0198027 log. cos, 9-,S99536 log. COS. 9 797563 Ead. COS. J = To find A B (e). -COS. fl.COS. c « = 40° « = 6i c=So 30' 4 29 log. log. log. sec COS COS oii8q55 9-684658 9S03613 ngle ABC, giver A B (c) 50 '40 ', A C (A) A ^ To find A. 8 Had COS A = t an. c . cot. b COS A=t in. c . cot. h c =■■ 50" 40 A = ii3 26 I So 00 A=i2i 56 log. tan. 0-08-47 1 log. cot. 9 636918 log. COS. 9-7233&9 Note. — In the triangle A B C, J the hypothenuse being greater than 90°. and c Ir"^^ than 90°, A is of unlike affection to C, or greater than 90". Also A being greater lli;iu 90° its opposite side a must also be greater than 90°. Quadrantal spherical triangles are also solved by Napier's rules re- versed : using the quadrantal side as the right angle, the angles Bdjacent to it, the complements of the other two sides, and of ti.u angle opposite to the quadrantal side, as circular parts. OiiLiQUE Spherics. Case I. Given two sides and an angle opposite to one of them, to find the angle opposite to the known side. As sin. of a given side : sin. of its opposite angle :; sin. of the other given side ; sin. of it.s opposite angle. To find the 3rd side. As sin. 1 diff. of the two known angles : sin. g their sum ;: tan. ^ diff. of the two kiio-wn sides : tttu. ^ the third side. Or, as cos. | diff. of the two known aiigl.- : COS. ^ their sum :: tjin. ^ snm of the two known si:Wi ; tiXD. i the third tide. SriTKKICAL TRI 59j Case II Given two angles and a side opposite to one of them, to tiud tbe side opposite to the known angle. As sin. of a given iingle : sin. of its oppiwitr side :: Bin. of the other given iingle : sin. of its opposite side. Ai «iu. i diff. of the two knov 4- dift'. of the two known angle t,. ^ tlie third angle. 5 1 ditr. of the two kn their sum sum nf the two knov Cases I. and II. may also he solved by drawing a great circle from the unknown angle perpendicular to the opposite side. This divides the triangle into two right-angled triangles. The segments of the divided side may then be found by right-angled spherics. In t 55° 3S'. phene triangle A B C, given A 84° 52', V,C or (a) 67° 5', and A (■•) To find the other parts. As sin. a : sin. A :: sin. c : sin. C. 'a^6j°~s' logrco.sec.^035706 A = 84 52 log. sin. 9-9yS255 c = S5 38 bg. sin. 9 9|t6S7 C =63 12 log. sin. 9-95a''48 From B draw a great circle B D perpendii aftWliou, both less than 90°, BD falls witliii liar to A C. Angles A and C being of like Ihe triangle. Then by Napier's rules : To find A C (h). liud. cos. C = cot. a . tan. D C tan. DC = c..s. C . tan. a a = 67°^' log. tan. 0373907 C = 63 12 log. COS. 9 654059 DC = 46S' log. tan. 10027966 A D = 7 27 *-U lit Kad. COS. A = cot. c . tan. A D ^tan. A D=cos. A tan. e ^ = 55° 3S'' A ^ 84 52 AD= 7 27 To find B. As sin. a : sin . A :: sin, b : s in. B. = 67° 5' log. cosec. 0035706 A = 84 52 log. sin. 9'998255 i = 54 iS log. sin. 9-909601 B = 6i 25 log. Bin. 994356» log. tan.o-.65o3^ log. cos. 8 951096 log. tan. 9 1 16729 If A and C are of unlike affection — i.e. one greater and one less than 90° — the perpendicular will fall without the triangle, and tiie difference between A D and D C must be taken to find h. This also will solve Case II., given two angles and a bide opposite to one of them, t« find the otheu- pnrtu C Cn.V IXTROnUCTION Caoe III. Given two sides C the included angle. C Let ABAC and the included angle A be given. From one of the unknown angles at C draw a great circle P^rpendicular to the opposite side. Then in the right-angled triangle A D C find AU it the perpendicular falls within the triangle subtract AD irom AB to find D B, and if the perpendicular falls without the triangle add A D lo A B, and the sum is B D. ' To find B C. As cos.of A D : COS. of B D :: cosof A C : cos of BO. To find the unknown angles. As sin. of side just found : sin. of the given angle : : sin. of either of the given sides : sin. of its opposite angU, Second Mtthod. To find \ sum of the unknown angles. As cos. 1 sum of the two given sides : cos. \ their diff. :: cot. i the included angle : Un. \ sum of unknown angles. fCoTB.-This '\ sum of the unknown angles is of the same name as the i sum of the sides. To find 1 diff. of the unknown angles. As sin. i sum of the two given sides : sin. 4 their diff. •: cot. i the included angle : tan. i diff. of the unknown angles. The i diff. being added to the \ sum will be the greater angle, and being subtracted from it will be the less. .-r -n /-. / n o.o ,-' .t.H A R li-\ In the spherical triangle ABC, given B 125° 36', B C (a) 81 17. and, A B (e) W° 13'. to find the other parts: c= 59°I3' « = 8i°i7' E= 125 36 <i=J|_.7 «=59__13 6248? To find angles C and A. ^=70° IJ- 2 5 = 62 48 log. eec. 047 "90 :i+J = 7o°i5' log. cosec. 0026329 log. COS. 9-991897 ^^^=11 2 2 leg. sin. 9-281897 log. cot.. 9710904 ?=e.2 4S ^«= 5 58 log. cot. 9-710904 2 A±C = 56 „ log. tan. 10173991 log. tan. 9019.JO A-C_ 5 58 2 5° «3C A + C 56 n 2 62 9 A. To fill 0= 50° 13' "= 59 13 r, = i25 36 »=..4 3S J bide fc. og. cosec. 1 14373 og. sin. 9-934048 og. sin. 99>oi4i log. sin. 9 95S:;-j SrilKUlCAL TKIGONOMKTRY. Glj Case IV. Given two angles and the included side. In the triangle ABC given angles B, C, and side B C, a: to find the other parts. Where two angles and an included side are given, a great circle may be drawn from one of the given angles per- jiendicular to the opposite side, and the angle BCD instead of the segment B D found. The difference between BCD and the given angle C will give A C D. Then To find the 3rd angle. As sin. BCD: sin. A C D : : cos. B : cos. A. If the perpendii'ular falls within the triangle the angles B and A are of the s:ime name ; if it falls without the triangle they are of different names. The Second Method is the same as in Case III., only for cota. of half included angle use tana, of half included side. Case V. Given the three sides of a spherical triangle, to find the three angles. Find the half-sum of the three sides. Take the difference between this half-sum and the side opposite to a required angle, then add together the log. cosecs. of the two sides containing the angle, the log. sines of the half-sum, and of the difference between the half-sum and the side opposite the required angle : Half the sum of these four logs will be the log. cos. of half the required angle. In the i-f.herical triangle A B C, given A B (<■) 79° 56', B C (a) 1 19° 36', and A C (i) t4° 5', 10 tiud angle B. lop. sin. square of .•7" 41' 20", suiple- ment B. 2 B = 52 18 40 Case VI. The three angles being known, to find a side. Add together the log. cosecs. of the two angles adjacent to the required side and the log. cosines of the balf-sum of the three angles and the difference between the half-sum and the angle opposite the required side. Half-sum of these four logs, will be the log. sine of Lalf the required side. <I = 119° 36' log. cose c. -060733 e = 79 56 log. cosec. 006738 i = f4 5 263 37 «3« 4830 log sin. , 987237? 67 43 30 log^ .99663.8, 19 906166 J B a" = 26 9 20 1 Og. COB. 9-9530S3 ( (52a INTRODUCTION. AMPLICATION OF THE PRECEDING CASES IN SPHERICAL TRIGONOMETRY TO Questions in Nautical Astronomy. The Amplitude. In these figures N ESW represents the horizon, S and N being its south and north points ; N Z S the celestial meridian ; the place of the body observed on the horizon, W the amplitude, P the pole of the heavens, P the polar distance, less or greater than 90°, as the declination of the body observed is of the same or of a different name to the latitude ; Z the zenith, W E the prime vertical, and W Q E the equator. From Right Spherics, p. 57a. In the problem to find the amplitude of a heavenly body, No. 884, there are given P N the lat. and P the polar distance to find W the amplitude. Then in right-angled triangle PON log. P N (lat.) + log. sin. | f ^ _ '^^, \ (dec.) = log. sin. ,;3jj. ie. W the amplitude. The question can also be solved by the quadrantal triangle Z P 0, where P Z 0, and therefrom W, may be found. Latitude fro.m Reduction to the Meridian. From Oblique Spherics, Case I., p. 58a. Given Z P the hour angle, P the polar distance, and Z the zenith distance, or two sides and an angle opposite to one of them, to find the remaining side P Z, or the colat. at the time of observation, see Nos. 7uO to 704 and explanation of Table 70, page 427. BPIIERICAI, TRIGOXOMETKT AND NAUTICAL ASTRONOMY. G3a The Hour Akule and Azimith. From Ohli(jiie 8[)heri-'s, Case V., p. 61a. Here are given : P /, the colat., P(J the pohir distance, and Z O the zenith distance, or the three sides of the triangle Z U P ; to find either Z P O the hour angle, or P Z the azimuth, see Nos. 614 and f.74. Iainars. From Oblique Spherics, Cases V. and III The Lunar problem is fully treated upon (see Nos. 836 to 863). The figures of 837 show the solution by oblique spherics, where first, in the triangle sZm, three sides, the two apparent altitudes Zm and Zs, and the aj)parent distance ma are given, to find angle m Z s ; and then in the triangle ]\I Z S, two sides, the two true altitudes ZM and ZS and the in- cluded angle Z are given, to find the true di.stance M S. Double Altitude. From Oblique Spherics, Cases III. and V., pp. 60a and 61a. For two altitudes of the same body the solution of this problem is fully given at No 757, and figure at p 2(18, where right spherics f, are used : seep. .'j7a. Ifdif- ^ I erent bodies are used, the ,-^ '' problem is solved by oblique spherics. Fig. 1 illustrates a double altitude where the observa- tions are taken of the same body and right spherics are used. In this case, A and B are the places of the body in the two observations ; PA, P 15, the polar distances ; 3 /Vy. 3. 64a INTHODUCnO!?. Z A, Z B the zenith distances ; A P B the polar angle or intervaL P D is drawn perpendicular to A B, dividing A P B into tvro equal parts ; Z F is drawn perpendicular to P D. Fig. 2 illustrates the problem where observations of two different bodies are taken, and the problem solved by oblique spherics. See No. 770, Note to pages 273, 274, and figure at page 268. Sumner's Method. From two altitudes of the same heavenly body taken at a requi- site interval apart, or two altitudes of different stars (having the requisite interval in azimuth) taken at the same time, two small circles (circles of position) may be described, the intersection of which * will be the place of the ship, allowing for her run in the Interval. In the figure A and B represent the j)laces of the body or bodies at the time or times of observation. From these points as centres, with the zenith distances, small circles are drawn, the intersection of which will be the zenith of the observer, or place of the ship. The intersection of these circles will be represented on the chart by the two straight lines C D and F Gr, drawn at right angles to Z A and Z B, the bearings of the body or bodies at the time of observa- tion. Full explanation of this useful method, with an illustrative chart, will be found under Nos. 1009 to 1014. Great Circle Saillng. From Oblique Spherics, Case III., p. 60a. Given P A and P B the two colats. and A P B the diff. long., to find A B the distance and A and B the courses from one place to the other ; or given two sides P A and P B, and included angle A P B, to find the other parts. The position of the vertex D will be found from the right- angled triangles A P D or B P D. This problem is fully treated upon in Nos. 336 to 347. » Achai'llcl showiiig this iutersoctiou will be fuuud iu Lei-k/o Wriukles, 'JtU eilit. j.i. 50a NAVIGATION. CHAPTER 1. Definitions. i78. By tlie ocneial term Navigation is meant that science • liich relates to the detennination of the phicc of a ship on the sea. 179. Tlio place of a ship is determined by either of two methods, which are independent of each other: 1st, by referring it to some other place, as a fixed point of land, or a former place of the ship herself; 2d, by astronomical observation. The first of these methods is treated under the head of Naviga. 7ION ; the second, under that of Nautical Astronomy. 180. The earth is nearly a globe or sphere : this is proved in three ways. 1st. When a vessel is seen at a considerable distance on the sea, in any part of the world, the hull is partly or entirely concealed by the water, though the masts are visible. 2d. 'J'he sliadow of the earth thrown on the moon when the earth is between the sun and the moon is, in all positions of the earth, circular. 3(1. The earth has been sailed round. The earth, however, is not exactly spherical, but of the figure called an oblate spheroid, which resembles an orange, the shortest diameter (that w hich joins the poles) being 7899 statute miles, and that of the fullest parts (about the equator) being nearly 26 more. 181. The earth turns once round in 24 hours. Tiie line round " liich if revolves, and which is the shortest diameter, is called thf iurij. and it^ exlronntifrs ar- llu- North and South Poles, as N, S- 56 NAVIGA-ilON. 1S2. The Eqcatoh, called also the Equinoctial Line, or vul^jarly the Line, is a circle equidistant from both poles, as W M E, and dividing the globe into two half globes, or hemispheres, N W E and SNVE. At all places on thia circie me sun nscs at 6 a.m., and sets at 6 P.M., all the year round ; the days and nights are thus equal, bein;j 12 hours each. 183. A MiiRiDiAN is a semicircle joining the two poles, aa N AS, N BS. Every portion of the meridian lies north and south, and places lying north and soulh of each other are said to be on the name neridian. 184. Latitude is the distance from the equator, measured on a meridian ; thus the latitude of a place A is A M, the latitude of IJ i.s BK. Latitude is named north or south, according as the place is north or south of the equator. Thus A is in north latitude, B is in south latitude. 185. The CoLATiTUDE is the complement of the latitude to 90°; thus N A, S B, N C, are the colatitudes of the places A, B, C. The colatitude reckoned from the other pole is the sum of the latitude and 90**; thus the colatitude of A is also S A, wliicli ii 90"+ M A (the latitude of A) : N B is the colatitude of B. 186. Latitude is measured in degrees, minnfes, and seconds. A minute, ov nautical viile, contains about 6082 feet, or 1013 fathoms, and therefore, a second is about 101 feet, or 17 fathoms nearly. See p. 104, note, and Spheroidal Tables, p. 724. 187. Circles parallel to the equator, that is, equidistant from it in every point, are parallels of latitude; as APH, b B. Two places in the same latitude are said to lie on the same parallel. 188. The Difference of Latitude ot two places is the portion of the meridian included between their jjarallels. Thus, A b is the dif- Jiprencc of latitude of the two places A, B ; C H is that between A and C. The difference of latitude of tlie ship is, therefore, the distance she makes good in a north and south direction. Difference of latitude is also called Nort.hing and Southing, and is marked N. or S. It is then said to be one of these names. 189. It is evident, that when two places are on the same side of the equator, their diff. lat. is found by subtracting the lesser latitmle fiom the greater; and that when they are on opposite sides of the equator, that is, when one place is in north latitude, and the other in south latitude, the ««/« of tiieir latitudes is tlieir diff. lat. Tlius the iliff. lat. of A and B, which is A b, is the sum of the north latitude A M, and the soulli latitude B K, or ]\[ b. X. I . Fin.l the diff. lit. of Cape Clear Cape Finisterre. Cape Clear 51° 16' N. and Ex . 2. Find (he diff. lat. of Cape V Cape St. Rocjue. CapeVerd 14° 4^ N- Cape St. R.»,«e s =8 S. Dirp. LAI. ao II Cape Finisforre .. 42 54 N. D.rr. LAT. 8 ja UErlMTlONS. 5" Ex. 3. A slii|) sails from lat. 50" 19' N. to | Ex. i. A ship sails fmni lat. T 11 N Ui +8 ' 12' N. : tint! Iicr iliff. lat. o" 13' S. : tiud lu-r ditl. lat I. at. left 50° 19' N. I Lat. left i" m'N. Lat. in 48 12 N. I Lat. in o 13 S. Di»F. LAT. 2 7 or 127 miles. DirF. lat. i 24 or 84 mile». Example» for Exercite. KcqMirecl the diff. lat. between the following places : 1. Between a place A in lat. 42° 21' N., and another 2. Between Halifax and the Cape of Good Hope. .\ns 4716 miles. 3. Between Diego Ramirez and Cape Lopatka. Ans. 6447 miles. 190. When a ship in north latitude sails north she evidently in f teases her latitude; and so, lii<e\vise, when in soutli latitude she siiils south; because, in these cases, she increases her distance from tlie equator, at which the latitude begins. But if in north latitude she sails south, or in south latitude site sails north, she diminishes her latitude. Hence, when one latitude and the difF. lat. are given, the other latitude is easily found. Ex. 1. A ship from 43° 3c' S. sails 219 miles south : required her lat. in. Lat. left 43° 30' S. Diff. Iat2i9' J 39 S. Lat. in 47 9 S. Ex. 2. A ship from lat. 43° 11' N. Ex. 3. A ship from lat. i" 3' N. s.iils iij miles south : required her lat. in. Lat. left 1° 3' N. Diff. lat. 123' _2 3^ S. Lat. i.v 1 o S. The ship being in 1° 3', or 63 miles N. o( .,-.,. • jv 1 . • the equator, must evidently be in S. lat. ,94 miles southing : required her lat. .n. ^^^^ ^^^.^^ ,^^ ^^^^^ ^^^4;^^ .^hus, ,r. Lat. left ......... 43 II N. 1 subtracting one of the quantities from th« Diff. lat. 194 ... J 14 S. oth^r, tlie difference takes the name of the Lat. in 39 57 N. I greater. Eaampletfur Exerciee. 1 . A ship from lat. 59' 27' S. sails southward until her diff. lat. is 374 : find lier present 2. Lat. left 48° 2' S diff. lat. 149 N. ; what is the lat. in ! Ans. 45° 33' S. 3. Lat. left 53^ 4' N. diff. lat. 122' N. ; find the lat. in. Ans. 55^ 6' N. 4. Lat. left 0° o', diff. lat. 2° 1 3' S. ; wliat is the lat. in .= Ans. 2° 13' S 191 . Longitude is the distance measured on the equator between Ihe meridian of a given place and another meridian, called the fnut meridian* The first meridian with us is the mcnoian of Green- wich Observatory; thus, if G be Greenwich (fig. in No. 180), the longitude of A is D M, the longitude of B is D K. The longitude of a place is named East or West, aeconling as it is to the east or west of the first meridian ; ihtis A is in west longitude, II is in east longitude. • The first meridian is a matter of arbitrary choice amongst different nations ; thui, tbf French refer to Paris. It is thexeforc neressary, in taking uv a chart, to observe what inei> iiaii the longitude is rrchon»l from. Sec y. 30.). 58 NAVIGATION. Il>2. We may use citlier the longitude of one name or tlie sup. piriuent to 36U°, with the contrary name; thus, instcatl of l()ti° W. ■we n'ay say 194" E. 19;i. IiOn<j;ituiJe is measurerl either ni space (or arc), that is, in degreus, minutes, and seconds; or in fAine, that is, in hours, minutes, and seconds, each hour being equal to 15 degrees ; for the sun. which regulates the time, returns to tiie same meridian again, after cluecribing a complete circle, or 360°, in 24 hours, and 15 x 24 is 360. 194. The DiFFERENCK OF Longitude of two places is the portion of the equator included between their meridians; thus MF is the difF. long, of A and C, as also of A and H, and of b and ('. To measure, therefore, the difF. long, of two places, we must follow down their meridians to the equator, and then take the included portion of the equator itself.* 195. When two places are on the same side of the first meridian, their ditf. long, is found by suhtracting the lesser longitude from the greater; thus the difF. long, of C and V, that is, the difference between D F and D K, is K F. But where the places are on opposite sides of the first meridian, that is, when one place is in east longitude and the other in west longitude, the sum of their longitudes is the difi'. long. ; thus the difF. long, of A and P, as also of A and B, is M K, which is the sum of M D and K D. When one longitude being east and the other west, the sum exceeds 180°, take the supplement to 360° for the difF. long. Find tlie diff. long, of Ushant and | Ex. 3. A ship sails from longitude 7° 56 the east point of Madeira. Ushant f 3' W. E. point of Madeira 16 39 W. Diff. long, ii 36 2. Find the diff. long, of tlie Cape of Good Hope and Tristan d'Acunha. Cape of Good Hope 18° 29' E. Tristan d'Acunlia ... 12 2 W. Diff. long. 30 31 W. to 18" 32' W.: find her diff. long. Long. left.. 7° 56' W. Long, in 18 32 W. Diff. lono. 10 36 A ship sails from longitude 1° f. to 2° 17' E. : find her diff. long. Long, left 1" 20' W. Long, in 2 17 E. Diff. lo.ng. 3 37 Examphsfor Exercise, Required tlie difference of longitude between tlie following places : 1. Between Halifax and the Cape of Good Mope. Afls. 4914 . 2. Between Ushant and St. Michael's. Ans. 1238'. 3. Between Diego Ramirez and C. Lopalka. Ans. 8071'. 4. Between New York and Manila. .\ns. 9899'. 196. When a ship in E. long, sails east, or in W. long, sail * Since the meridians are all parallel at the equator and meet at the poles, the distance between any two meridians, measured east and west, is less as the latitude is greater ; that i», the absolute number of miles, or of feet, in a degree of longitude, is less as the Latitude in which tliey are measured is greater. Hence, also, a given number of miles between two meridians corresponds to a greater diff. long, as the latitude in which they are measured ii greater. For e.xamjile, two places in lat. 10' and distant •40 miles east and west from each other, hiivp 40-fi ditr. lonj. In lat. .iO ' two places similarly situated have 1° 2'-2 diff. long, ttucitions 1)1 tliis kind iiie solved bv llie rules of I'ai.dlcl Sailing. 5» she f vidoiitly increases lier loiii,Mtiii!c, oi- the distiinee from the first meridian. But if in E. long, she sails west, or in W . long, siie sail? east, she diminishes her longitude. Hence, when one longitude in given, and also the ditf. long., the other longitude is easily found. Ex. .■?. A ship from long. o° 32' W nmliri 7.^ 8' tasting : find the long. in. Long, left o" 32' W. Uilf. long 2 8 E. Long, in 1 36 E. Ex. 4. A ship from long. 178^ 54' Vr nidUus 3° 4' westing : find the long. in. Long, left 178° 54' W. Diff. long 3 4 W. Long, in iSi 58 W. Or (by No. 195) 178 2 E. Examplen for Exercise. 1. Long, left 1° 25' W. diif. of long. 85' E : wliat is thelong. in? Ans. 0° o 2. Long, left 0° o', diff. of long. 146' W : tlic long, in is required. 3. Long, left o^ o', diff of long. 122' E : what is tlie long, in ? 4. Long, left i6o°2o'W. diff. of long 4i°2o' W: find the long. in. 5. Long. left. 179° lo' E. diff. of long. 84' E. : what is the long, in .' Ex 1. A ship from long. 51° 40' E. sails east 3^9': find the long. in. Long, left 31" 40' E. Diff. long 3 9 E. Long, in 34 49 F. K.X 2. A ship from long. 07° 45' \V. makes ]" 11' easting; find the long. in. Long, left 97° 45' W. Diff. long I II E. Long. ,n 96 34 W. Ans. ' i6' W. Ans. 2° i' E. Ans. i58°»o' E. Ans. 179° 26' W. 197. The Course steered is the angle between tlie meridian and th(! ship's head. The course wade good is the angle between th". meridian and the ship's real track on the surface of tlie sphere. The course is reckoned from the north, towards the east or west, when the ship's head is less than eight points from tlie north jjoint. The same applies to the south point. The course is measured in foints of 1 1° 15' each, or in degrees and minutes. 198. The track of the ship while preserving the same angle with all the meridians as she crosses them in succession, is called the Rhumb Line. 109. The Distance between two places, or the distance run by the ship on a certain course, is measured in nautical miles of (JO to the degree of latitude. See p. 104, note, and Table 64 A. Three such miles make a nautical league. 200. Tiie DEPARTirRE is the distance in nautical miles, made good by the ship due ea-st or west; or the distance between two places measured along their parallel. Departure is marked east or west, according as it is made good towards the east or west, and is accordingly called easting and west- ing; such easting and westing being, however, exjn-essed in miles, and not, like longitude, in arc. Thus, if a ship sails from a place A to another as B, A B is the tiie HHirlc 80 NAVIliATlON. CAB the course ; B C drawn E. and W., or perjwndicular to C A, is the dejiarture; and A C is tlie dijf. kit. 201. The Bearing of an object or place is the angle contained between the meridian and the direction of the object, and is the same thing as the course towards it. Taking- a bearing of an object is called setting it. The bearings of two objects, taken from the same place, consti- tnte cross bearings, the lines of direction of the two objects intersecting or crossing each otlier at the place of the observer. 20"i. Leeway is the angle included between the direction of the ship's keel and the direction of tiie wake she leaves on the surface ol the water. Tluis the vessel C, while she moves tlirougii the water in the direction of her length, in the line C B, is at the same time pressed to leeward of this line by the force of the wind, supposed in the figures to blow on the vessel's left or port side ; her wake, or actual path through the water, appears therefore to windward of the line which she endeavours to keep, as is represented by the line C L. The angle A C L is the leeway. The course steered (No. 197) is the angle N C B, N C being the meridian ; the course made good is N C D, the line C D being dc'tci mined by producing LC. 203. Tlie Dead Reckoning is the account kept of the ship's ])la<-p, without reference to astronomical observation. It is written U. K. for shortness. 204. The Visible, or Sea Horizon, is the apparent boundary of the surface of the water, which appears to the eye the circumference of a circle. 205. The Depression, or, as it is called by abbreviation, Dip, is the angle through which the sea horizon appears depressed, in con- He([uence of the elevation of the spectator. Suppose the spectator at A, above the sea, and A H a line perpendicular to the plumb-line at A, which tends to the centre; A H is the true level, or horizontal line, and the angle H A B, in- eluded between it and the line A B, toiuiiing the .sea, is the dip. DKFIN1T10>S, 61 Tlio (lip (lepoiuls on the distance in naiiticiil miles of tlie visible huil/.on. Tims, to the eye 30 feet above the sea tlie true tlij) is 6', or Jlie distance of the iiorizoii itself is about (i miles. This is easily proved tluis, Let C be the centre of the earth, the place of the observer ; then the line OB drawn touchin<5 the surface at. B determines B the farthest point visible to him. Draw OH perpendicular to O C, then ginee O B touches the circle af B, the angle CBO is a riu-ht anijle (No. 138, Cor.) Hence UCA is thewmplenient of C O B, and HOB is also the complement of COB (C0H=9()°), therefore A C B and HOB are equal. The depression is given in Table 8. 20G. The Altitude of a terrestrial or celes- tial object above the sea horizon is the angle included between the line drawn from the eye to the object, and the line from the eye to the horizon. Thus, the angle M A B is the altitude of the summit M. The altitude here, in consequence of the great elevation of the spectator at A, about ^ of the radius, or 330 miles, is less than tho dip, or the summit M is really below the true horizontal line A H. This mav take place when, from the small height of the object with respect to that of the observer, or its great distance, it is seen very little elevated ; but in most eases A M will fall above A H. 207. The rays of light which pass from any distant object on the earth suffer a change in their direction, which is called the terrestrhd refraction, by which the object appears in general higher than its true place. This effect is, on the average, about i'4 of the intercepted arc, or distance in miles, which are minutes of a degree very nearly. Thus, an object twenty-eight miles distant is raised about 2' above its true place. The sea horizon is thus raised by refraction, or the apparent dip (Table 30) is less than the true. This proportion, however, is subject to great irregularity, and varies between \ and ^ of the intercepted arc. The apparent eleva- tions of the summits of high land are thus subject to great variations, depending on particular states of the air. 208. The apparent place of the sea horizon differs also in differ- ent temjieratures of the sea and air. When the sea is warmer than the air, the horizon appears below its mean place, or that at which it appears when the air and water are of the same temperature, or the apparent dip is too small; when the sea is colder than the air, the horizon ap]>ears above its mean place,* or the apparent dip is too great. * Admiral W. F. W. Owen informs me that he found on one ocnasion, in observing ii star's •Ititudc, a change of 4' in the plare of the sea horizon, in the tropics, soon after sunset. Mr, Kisher observed a variation in the place of the horizon of 18' in the arctic regions. In summer the ice horizon was elevated, not dcjiressed ; in the winter it was depressed several minuti s. — {AppemlU 111 Cajitaia Parry's Voi/aye in 1821-3, p. 1 87.) Tlicse observations, how«v< r, ilo «ot nil folbnv the rule above. A table for correcting the r.ppaniit place of the sea horiZHii for the difli iince of ti nipciaturc ol 'he sea and the air, acoordini; tu the hcii^ht of the eve, wuiihl 62 NAVIGATION. Colonel Sahine '''fes a table of depressions observed from the gangway of H.M.S. Phcat-ant, at ISfr. liii. above the sea, in the Gulf Stream, ami after kaviiig it.* On Dec. 5, 182.', lat. 36°i N., long. 72°A W., at 10'' a.m., the temperature of the sea being "0°, that of the air 60°, the dip observed by WoUaston's dip sector was 4' 57", or 1' 6'" more than the lalde. At noon ihe temperature of the water had clianged to 62° 4, the air at 60° as beibre, the ship having passed from the warmer water of the .stream to the colder water of tlie rest of the "Cean, ani the dip observed was 3' 37". Fioin the result of iiis ol»ei vat oiis, Cjlonel Saliine considers that the navigator will be right nine times in ten in assuming that, wlien tlie sea is warmer than the air, the tabular dip is too small. In only one case, however, did this error ever amount to so much as 1' :>6", the sea being then at 49°, and the air at 38°, or the difference 1 1° ; and it is important to remark that tlie error of the table is by no means pr.'portional to the diftlreuce of these temperatures, which in one case was no less than 2!)°. Numerous instances are on record, in the accounts of modern navigation, of errors of observation arising from variation in the pUce of tlie sea horizon. 209. Besides the -vertical effect of refraction above described, gome instances have been recorded of a sensible change in the hori- zontal direction of objects. Mr. K. B. Martin observed a change in the true direction of a point of land in the Azores, towards sunset. He also mentions an extraordinary change in the direction of C. Grisuez light as seen from Eamsgate at the close of a very hot day ; on which occasion, also, distant objects were elongated horizon- tally till they seemed to separate into parts. ("Naut. Mag." 1847.) Lieutenant Wilke< ohserved from the summit of Mowna Ron, the sun's horizontal diiimcter 1 ngilnned ont lo twice and a half the vertical one. ('-Narrative of the United .Stare'. lA|il.irin^' I Expedition," 1838-42 ) In the Survey ot the Isthmus of Tehuante|iec, unl. r St h.;r Ci. Miro, in 1842-3, the refractions at San Mateo on tlie Pacific, " especially the lateral (Mies," pioduced the strangest illusions.f 210. The Tkopics of Cancer and Cafeicorn are the parallels of latitude 23° 28' N. and S. These are the dotted lines nearest the equator (6g. in p. 55). The sun is vertical at noon twice in the year to every place between the tropics, and never to any place outside of them. The space between the tropics is called the Torrid Zone, on account of its heat. 211. The Arctic Circle, or Nortli Polar Circle, and the Ant- arctic Circle, or South Polar Circle, are parallels distant 23° 28' from each pole, and are therefore in latitude 66° 32'. These are the dotted lines nearest the pole. Within these circles the sun does not set during part of the summer, nor rise during part of the winter. The spaces within these circles are called the Frigid Zones, on account of the cold. The spaces between the tropics and the polar circles are called the Temperate Zones. be useful ; but there arc scarcely any data fur the constrmtion of such a table, and the theory itself appears not to be complete. The above variation of the place of the apparent horizon, with mirage, reflected images, and other optical illusions, were tirst discussed, generally as ip'estions of unequal tempera- ture alone, by M. Biot, Mem. de I'lnstitut, 181)9. • Account ofExperiments to determine the Figureofthe Earth. London, 1825,p.4.54. t It is easy to conceive, that if a mass ol air of ditferent density from the rest he inter- posed between the spectator and the object, and i( also the sides or faces which he looks Ibrougb be not exactly parallel, it will have the effeet of a prism, and will seem to throw the o'lject to the right or left of its true direction. If the -nrlaccs are curved, the cHVcl ut uiu^inliing or diuiiuishiiii^ will occur at the tame tinie. CHAPTER II. Instruments of Navigation. I. The Compass. II. The Loo and Glasses. The necessary instruments of navigation are the Compass, hj the aid of which the course of the ship can be directed ; and the Log, which, with the help of sand-glasses for measuring small intervals of time, or a watch showing seconds, gives the velocity or rate of the ship, and thence the distance run in any interval of time. I. The Compass. 212. Before the invention of the Compass, the course; of the ship was directed by reference to the land, or to the position of the heavenly bodies ; but when those objects were obscured, the Beanian must sometimes have been much perplexed. The pointing or directive property of the magnet, on which the efficiency of the compass mainly depends, appears to have been known to the Chinese, and made use of by them in travel- ling by land and sea, in times of remote antiquity. The an- cient Greeks and Romans, though familiar with the magnet, were not apparently aware of its directive property, nor were their descendants till the beginning of the thirteenth century. About that time the seamen of the Mediterranean gradually became acquainted with the fact, that a piece of magnetised steel, shaped like and commonly called a needle, would, if allowed to turn freely about its centre, always come to rest in the same direction, and that, by reference to its pointing, they could roughly check or direct the course of the vessel. Thus, before the seamen of those days were two problems. First, the best means of giving to the needle freedom, to take up any horizontal direction, and of indicating the direction of the ship's head relative thereto. Second, In find the exact direction of the pointing of the needle, in relation to some known standard of direction. In other words, first the per- fecting of the mariner's compass ; second, a knowledge of what 13 now called its variation. Appai-ently, the earliest means used to allow the needle to take up any position in azimuth, was by thrusting it through ft piece of light wood or pith, forming with it a rectangular ciusiii 64 NAVIGATION. the wood or pitli being just sufficiently large to float the neeille, when the cross was placed in a vessel of water. Otherwise, the needle was poised at its centre on a sharp pivot, and inclosed in some form of box. Subsequently, the necessity for keeping the box horizontal, in the varying motion of the ship, was met by gimballing the compass-lox, and, for convenience, a circular disc of paper, called the fly, having a graduated circumference, was placed on the needle. The fly and the needle together was calK'd the card. The box was generally made of brass, shaped like a basin, and had a glass cover. A mark, called the lubber-line, was placed on the fore part of the compass-box, or bowl as it was commonly called, on the inside, indicating the direction of the ship's fore-and-aft line, from the centre of the card. The circumference of the card was divided into thirty- two divisions, called points ; these were subdivided into half- points and quarter-points. The four principal points, or, as they ai-e called, the cardinal points, are the North, South, East, and West ; the East being towards the right when facing the North. All the points of the compass are called by names composed of these four terms. The points half-way between two cardinal points are called after both of these points : they are the north-east (written N.E.) ; north-west (N.W.) ; south-east (S.E.) ; and south-west (S.W.). These points are sometimes called quadrantal points. A point half-way between one of these last and a cardinal point is called, in like manner, by a name composed of the nearest cardinal point and the adjacent point, N.E., N.W., S.E., or S.W. Thus the point between N. and N.E. is called north-north-east (written N.N.E.) ; the point between E. and N.E. is called eaatr noith-east (written E.N.E.); and ao of others. rXSTRUMEXTS OY NAVIGATION. fio ITie points *iext the eight principal points (namely, N., S., E., W., and N.E., N.W., S.E., and S.W.) take the word bif between the name of sucli point and the next cardinal point. Thus the. point ncA.-t to north, on the east side, is called North hij East; that on the West side is called North bij West. Thus, on in- specting the compass, it is easy to see the reason of the names E. by N., S.W. by W., &c. A half-point, which is the middle division between two points, is called after that one of its adjacent points which is either a cardinal point, or is the nearest to a cardinal point. Thus the middle division between N. and N. by E. is called uortli-/ia//-east (written N. ^ E.). Half-points near N.E., N.W., S.E., and S.W., take their name from these points. Thus we say N.E. ^ N., and N.E. i E., and N.E. by E. ^ E. The same holds for a quarter and for thi-ee-quarters as for a half-point. In speaking of these divisions of the card, brevity seems to have been the chief end, rather than the habitual reading of the card from left to right, or the reverse. Thus, we may say N.E. by E. ^ E. ; but continuing to the right, instead of E.N.E. ^ E. and E. by N. i E., it is usual to say E. by N. ^ N. and E. ^ N. The name of the opposite point to any proposed point is known at once, without referring to the compass, by sim.ply reversing the names or the letters which compose it. Thus the opposite of N. being S., and that of E. being W., the opposite point to S.W. by S. is at once known to be N.E. by N. The opposite of W. | S. is E. I N., and so on. Dividing the circumference of the card, by successive halving, into points, half-points, and quarter-points, was well adapted to the time, not very distant, when many helmsmen were unable to read. The quarter-point was also considered tlie smallest division a man, sometimes under the blinding influences of wind, rain, and spray, could well distinguish. Now, however, the cards of steering compasses are frequently divided to degrees, in addition and external to the point divisions. In cards of nine or ten inches in diameter, the degrees are suificiently large to be distinguished by men of ordinary sight. The degrees are always marked from North or South, towards the East or West ; tlie courses, therefore, are read from left to right, and vice versa, in alternate quadrants. This is apt to cause mistakes in steering. For this reason, and for precision and brevity in speaking, writing, and signalling, there is much to be said in favour ni marking the card from zero to 360 degrees, round by the right. Small compasses for shore work are thus marked generally. Repeating the points in any order is called boxing tlie com- p(tKs; to do this is, of course, one of the first things a seaman learns. In becoming familiar with tin' points of the comp!i«s tha m NAVIGATION. learner should bear in mind that their utility is far from being confined exclusively to navigation, and that in finding hii way across a new countiy, or through the streets of a strange city, no impressions will be so distinct or so permanent as those grounded on the points of the compass. 213. As the ship's course, which is sometimes expressed in points and sometimes in degrees, is always reckoned from the north or south point, the seaman has to refer at once, in using the Tables, to the number of points, or degrees, in any course given by name. The following table, whii-h exhibits the degrees, minutes, and seconds, in each quarter-point uf the cunipass, will be con- venient for reference : — N— £ N-W S- 10 S— W Vi-. Nurth. Nurth. Hdiith. South. NJE N^W SiE SiW i N^E NiW SiE SiW \ N|E Nf W SaE sjw NbK Nb W SbE SbW I N b !•: 1 E N b W 1 W SbEiE S b W } w 1} N b I-: i E NbWi W S b E i E S b W i w l\ N b E i E NbWa W . SbEJE S b W f w •f NNE NNW SSE SSW 2 NNEiE NNW J W SSE \ E SSWiW 2\ NNE 1 E NNW i W SSE 1 E SSW i w 2l NNEJE NNW a W SSEJE SSWJW 23 NEbN NWbN SEbS SWbS 3 NEiN NW f N SE-JS SWJS 3} N K 4 N NW^N SEAS SWIS 3.^ N£iN NWJN SE JS SWiS 31 NE NW SE SW 4 NEiE NW JW SEJE sw \ w 4f NEiE NWiW SEJE SWlW 4l NEfE NWJ W SE3E SAVaw 4| SEbE NW b W SEbE SWbW 5 NEbEiE NWbWiW SE b E i i: SW b W 1 w Sk NEbEJE NW b AV i W SEbEiE sw b w i w si NEbEJE NWbWaw SEbE^E sw b W 3 w Si ENE WNW ESE wsw 6 E b N } N WbNf N E b S 1 S W b s f s 6} E b N i N W b N i N EbSiS W b S 1 s 6', EbNiN W b N i N EbSiS W b .S i s 63 Eb N WbN EbS AVbS 7 EiN WJN EJS Wi.S 7i EiN WIN EiS AV i S 7.V EiN W i N EJS W i s 74 Ea.vt. WeM. E.ist. West. S 2 4S 45 5 37 30 S 26 '5 M 15 14 3 45 l6 52 30 19 41 'S 22 30 25 IS 45 28 7 30 56 '5 33 45 36 33 45 39 22 30 42 II 15 45 47 48 45 50 37 30 53 26 15 56 .5 59 3 45 61 52 30 64 41 15 67 30 70 iS 45 73 7 30 75 56 IS 7845 Si 33 4S 84 22 30 87 n '5 90 2H. The Azimuth Compass is a compass of superior tonstri tion, especially adapted for observing bearings. It is titled \vi TNSTRUMENTS OF NAVIGATION. 07 two vortical viiiies. The one near the eye in obsei-vinrr, hiis ;i ntiiTOw vertical slit, with coloured shades (or observing the sui;. The other Viuie has a wider slit or opening, having a vertical thread in the middle of it. In front of this vane is a reflector, for observing objects elevated above the horizon. The line join- ing the slit in one vane, and the vertical thread in the other, should pass over the centre of the card. The cards of azimuth compasses are always marked to degrees, and frequently to anialler divisions. In the Prismatic Azimuth Compass, a magnified image of the divisions of the card is read by reflection, in a prism attached to the fore side of the near sight vane. Azimuth compasses being required for taking bearings, are placed on a tripod for shore work, and on an elevated stand on board ship. 215. In the early part of the present century, when ships and instruments for navigation were rapidly improving, the compass was still a rude instrument, and not abreast of the requirements of the seaman. In 1820 Mr. Barlow reported to the Admiralty, that half the compasses he had at their request examined, be- longing to the Royal Navy, were useless. It is probable that the compasses of the Mercantile Navy were no better. In 1837 their Lordships appointed a committee to inquire into the matter, and, if possible, to find a remedy for an evil so pregnant, as they said, with mischief. This step was taken for the benefit of the Royal Navy, and the improvement which took place, both in the design and in the workmanship of the compass, in consequence of the recommendations of the Admiralty compass committee, was of immediate and lasting benefit to the public service. The Mercan- tile Navy was not so immediately benefited, as the proceedings of that committee were not made public. But doubtless the fact of there having been such a committee stimulated compass makers to seek information, and to apply it to the improvement of the mariner's compass. A great difficulty to be overcome, in a compass intended to be used on board ship, is the disturbance of the card caused by the motion of the ship. The Admiralty compass committee, while insisting on extx-eme lightness in the fly and fittings of the card, made considerable addition to its weight, by applying more needle power than would otherwise have been desirable, in order to secure steadiness. This was a fairly successful way of meeting the motion of ships at that date. But the violent and continuous motion, subsequently caused by the general adoption of the screw propeller, has been generally met, by suspending the compass bowl by springs or india-rubber. The difficulty of getting a compass that would be steady in small vessels and boats, led to the introduction of the Liquid Compass; that is, a compass having the bowl filled with liijuid instead of air. Tin; fir.st practiital liquid compass was patented 68 NAVIOATTOW. by Mr. Crowe in 1813. It was eubseqiiont-ly improved by other makers, and is now, when well made, a very efficient compass for all purposes. It is especially adapted to stand severe vibration, and the shock of gun-firing. For these purposes, and for use in boats, it has not yet been excelled. 216. In 1876 Sir Wm. Thomson patented a compass, which la regarded with much favour by navigators. At the circumference of the card is an aluminium ring ; the cap is held in the centre by radial silk tlu-eads, extending from it to the ring. Attached to the ring and threads is a disc of very light paper, its circumference having the usual compass graduations. All the central part of this disc is removed, still further to lessen the weight. Recognis- ing the fact, that the power of a magnet increases relative to its weight, as the size decreases, the needles are very small. They are suspended under the card from its circumference. The entire card is not more than one-fifth to one-tenth of the weight of compass cards generally, of the same size. The friction on the pivot is, therefore, proportionally diminished. By giving to the card no more needle power than would certainly overcome this much-diminished friction, it has a very slow period of vibration. The desirability of giving to a compass card a period of vibration that would not be isochronous with the roll of the ship, in order to maintain steadiness in a seaway, had already been pointed out by Mr. A. Smith, and by Mr. Towson. The bowl is protected from disturbance, also, by being suspended on a twisted wire gromet. This compass card, from the little friction on the pivot, is very sensitive at all times. From its slow period of vibration, it is steady when the ship is rolling ; and, by reason of the suspension of the bowl, it has considerable immunity from the disturbances caused by vibration, shakes, and sudden shocks. 217. Though a compass, when supjilied to a ship, should bo accurate and efficient, it is desirable that the seaman should be able to satisfy himself on these points. The following essentials should be looked to, in steering and azimuth compasses, as far as they apply to each kind respectively. The point of the pivot should always be in the same plane as the centre of the gimbals. The pivot should be sharp, or, when intended to be a little rounded, quite smooth ; it should be free from rust. The cap should be sound — that is, not cracked nor perforated — and free from dust or dirt, which sometimes gets into it. Placing the card gently on the pivot, it should be deflected two or three times, through a small angle from its position of rest, to see if it always comes back to rest at the same point. This would show if the needle power is sufficient to overcome the friction on the pivot. 'Select a position on shore, free from disturbances, from whence the bearing of some object is known. Measure horizontal angles INSTUUMENTS OF NAVIGATION. 69 fi-oiii it with a sextant, or other means, to three other objects, so selected that the correct bearing of four objects, about 90'^ apart, may thus be known. Now turn the compass round horizontally, so that the line from the centre of the card to the lubber-line coincides, in horizontal direction, with the line from the centre of the card to each object in succession. At each position of the compass, observe the bearing of the first object, by the siglit vanes. Assuming that the card is regularly divided, these observa- tions would show whether or not a course shaped, or a horizontal bearing taken, by the compass is correct. Placing the conipiiss on board ship in its binnacle, see that the bowl takes up its proper horizontal position in the gimbals ; that the lubber-line is vertical, and that a line from the centre of the card to the lubber-line is exactly in tlie same horizontal dii-ection as the fore-and-aft line of the ship. See that the thread in the sight vane is vertical, by testing it with a lilumb line ; and raise and lower the reflector, and see that the reflected image of the thi-ead coincides with the thread itself. This will show that the bearing of an object at any elevation, whether taken by direct bearing or by reflection, is correct. Metal pivots become blunted by wear, and steel pivots are also very liable to rust; jewelled caps naturally get worn and perforated by use, especially from the long-continued working of the screw propeller. They are also liable to be cracked by sudden concussion. Heavy cards are sometimes fitted with speculum metal caps, and work on jewelled pivots. Defective caps and pivots are a fruitful source of inefficiency in compasses, and require the esjjecial attention of the navigator. 218. At a time when ships had no compass in an elevated position, all bearings had to be taken from the steering compasses. These were low down to the deck, and therefore inconvenient tor that purpose. And subsequently, when most ships had an elevated compass, its position was frequently such, that an all- round view could not be obtained therefrom. The difficulty was met by the introduction of an instrument called a dumb card, or bearing-plate. It consists of a circular plate of metal, graduated like a compass card, and so gimballed that it may be revolved round a central pivot, in a horizontal plane. Adjacent to the circumference is a mark, similar to the lubber-line of the compass. It is fitted with sight vanes, shades, and reflector, for taking bearings. Tlie instrument may be placed in any position from whence the object, or objects, to be observed may be seen. The greatest care must be taken to see that the line from the centre; of the bearing-plate to its lubber-mark is in the exact fore-and- aft line of the ship. This may be 4one by referring it to some mark in the ship, exjictly in the fnre-and-aft line; or to snme mark, such as a IjoUard, which, from the position chosen for the 70 NAVIGATION. bearing-plate, is a known, small, and constant angle from tlie foie-and-aft line. If the direction of the ship's head by the hearing-plate, be made to correspond with the direction of the ship's head by any compass, then the bearings taken by the bearing-plate will be the same as if they were taken by that compass. And, con- versely, if the bearing-plate be turned round, so that the bearing of an object by it corresponds with its known correct bearing, the direction of the ship's head, as shown by the bearing-plate, is correct. This instrument, sometimes called a Pelorus, is exten- sively used. Another instrument, called a Palinurus, is sometimes used for getting true bearings. It is, simply, the mechanical con- struction of the celestial sphere, with its great circles. By means of time, latitude, and declination of some heavenly body, a line in the instrument may be set to the true direction of that body. All tlie parts of the instrument, when that line is pointed to the body, will be in the true astronomical direction, and the bearings on the horizontal circle of the instrument will be true bearings round the horizon. A mark placed as the lubber-line will, of course, show the true direction of the ship's head. It will be seen that, with this instrument, no calculations or azimuth tables are required to get a true direction. With respect to the use of such adjuncts to the compass, as have been briefly described, liability to secondary errors, both personal and instrumental, must be taken into account. To work directly, from a well-placed standard compass, appears by far the safest practice in navigation. Variation of the Compass. 219. The second problem before the early navigators was, to find the direction in which the needle pointed (No. 214). When the directive property of the magnet was first brought into use by seamen, it is probable that they continued for some time to steer by the sun and stars, as before. It was only when those objects were obscured, that they had recourse to a rude form of compass, to enable them to maintain their course, till their accustomed and more reliable guides appeared again. What the compass needle was to the seamen of those days, it is to the navigator of to-day. By it he can preserve a course, without reference to the heavenly bodies, for a longer or shorter time, and with more or less accu- racy, according to the perfection of his compass, and to tlie degree in which he is acquainted with the laws which govern its pointing. The natural standard of direction is the meridian. The hori- zontal angle contained between the direction of the meridiiui rSSTHUMKNTS OF NAVIGATION. ( I and the direction of tlie needle, is called the Variation of tlie L'ompass. It is termed easterly or westerly, according to whicli 6i(le of the meridian the north end of the needle points. The approximate direction of the meridian was easily seen in the northern hemisphere, by the position of the pole star. It must, therefore, have been well known, to all who noted tlio pointing of the compass needle, with any degree of care, that its direction did not coincide with the direction of the meridian ; or, in other words, that it did not, in all places, point to the nortli. This fact seems to have been brought most prominently into notice by Columbus. He found, on his first voyaue, in 14'J2, when well over towards the West Indies, that tbe needle pointed to the westward of north. In the seas which Columbus had hitherto navigated, as far as can be now judged, it pointed to the eastward of north. At the port in Europe from which he sailed the variation was, apparently, not less than two points easterly. Pro- bably, therefore, it was the change, and especially its going from easterly to westerly, rather than the existence of variation, whicli arrested the attention of Columbus. The first good determination of the variation, in England, was made in 1580, when the direction of the north end of the needle was about one point to the eastward of the meridian. Since that time, the variation has been observed with increasing frequency and accuracy. The following is an outline of the change in the variation in England. Commencing in 1580 at 11° 15' easterly, the north point of the needle moved towards the meridian, and crossed it in 1(>57, moving westward at the i-ate of 10' annually. The north end of the needle continued to move westward, with a diminishing ratf, till 1818, when it attained the limit of its western range, 24° ^58' westerly. Since that date the north point of the needle has moved to the ea.st with an increasing rate. The variation in London is now 17° :10' westerly, diminishing at the rate of 8' annually. The first attempt to give a comprehensive view of the direc- tion of the compass needle, in all parts of the world, was made by Halley, in a chart published in 1 700. This chart embraced the results of a voyage made by Halley himself, and such other information as was at that time available. Joining, by a line, the points on the earth's surface where the variation was the same, he traced, on a Mercator's chart, a series of lines of equal variation, extending over the Atlantic and Indian Oceans, and as far east as the meridian of 150°. Sevei-al similar charts, more complete and accurate, as the materials for compiling them increased in quantity and value, have since been published. The latest variation chart published by the Admiralty is all that the seaman can desire. On it the annual change of variation is bIso shown, enabling the navigator to obtain the variation very ckbt.'ly, at auy date subsequent to that of the publication of the 72 NAVIGATION. clmrt. CompariTig Halley's chart with those which have since been made, it appears that changes iu the variation, analogous to those observed in England, but of greater or lesser extent, have been going on nearly all over the world. The variation of the compass is thus shown to be a variable quantity, changing at a variable rate. Such being the case, the only way in which it is possible to make and maintain an accurate variation chart, is by the co-operation of navigators, in making and recording, for that purpose, observations of the variation of the compass, in all those parts of the woi-ld over Avhich they may sail. 220. Besides the change in the variation, which reaches its limits in long intervals of time, and is called the secular change, there are smaller changes, called periodical. Such is the diurnal change, wherein the needle moves through a small angle to the westward during the day, and retm-ns to the eastward during the night, in the northern hemisphere. In the southeru hemisphere, a similar change takes place, but in an oj)posite direction. The needle is also disturbed by the aurora, and by phenomena called magnetic storms. These changes are, iu the navigable parts of the globe, too small to be of any importance to the navigator. Neither is the pointing of the compass needle affected by atmospheric phenomena, such as fogs, rain, wind, or thundei'storms. Bui in caseu where a ship has been struck by lightning, the directive property of the compass needle has sometimes been impaired or destroyed. There is, however, one cause of disturbance of the needle which should interest the navigator. Humboldt, in the beginning of this century, observed that the needle, in certain places on land, was deflected from what may be called its normal direction, by some property in the ground. In previous editions of this work, several places are noted, where the variation was affected l)y the land, or by the ground in shallow water.* It is probable, from the practice of steering by the land when it is in sight, rather than by compass courses, that this distui-bance of the compass needle has escaped notice in some places whei-e it exists. It is, therefore, desirable that this unquestionable source of danger should be pointed out, that the seaman may be on his f^uard, when navigating near the land, or in shallow water, espe- cially in volcanic regions. Methods of determining the variation of the compass are given in Chapter VIII. 221. To correct compass coiirses and bearings for variation. The manner of doing this appears thus. Suppose one compass card to be placed directly over another, and the lower one to be irue. Now suppose the north point of the upper compass to be drawn two points to the right of the true by easterly variation, then the North point of the upper or muynetic compass corresponds • Comtnunder W. U. Moore of H.M.S. Perguin ^c. llic iicecilr. (55°) in 9 fuliunis, 2 miles Iruiu the shoix',, 11 euabt of AuhtiHlia. int Notice to MarinsrH, No. lo of 1 INSTRUMENTS l>F NAVIGATION. 73 to N.N.F. of the true compass, which point is to the riu^ht of N., and the South point corresponds to S.S.W. of the true compass, to the rif^ht of S., and so on. The contrary wo aid take phice v.ith westerly variation; hence to correct a magnetic course or bearing we have this rule. Rule. When the variation is easterly, apply it to the rvjlit of the compass course or bearing ; when wederhj, ap[>ly it lo thi; left, looking from the centre of the card over the point to be corrected. Ex. 1. Course by compass, S. J W.; fariation, 2j points easterly. Tkub Couhse, 2|^ points to tlie rifrlit of 6. i W., or S. 3 points W., or S.W. by S. Ex. 2. Course by compass, N. by E. ; T;iri.itiiin, 2 point westerly. 'I'lirK Course, 2 point.s to the left of N. by E., that is, N. by W. | To reduce a true course or bearing to the compass course or bearing, apply the variation the contrary way to that directed iibove. Ex. 3. Course or l)pariii^ by comj'; N. 84° E. ; variation, 19° W. TuuB CouusE, N. 65° E. Ex. 4. Ci'urse by compass, S. 4° ,'ariation, 17° E. Teue Couese, S. 13° \V. Ex 1. True course, N.E. by E.; varia in, I point easterly. Course by Compass, N.E. Ex. 2. True course, E. ^ N. ; variation ConiisK i)V Compass, E. by S. Ex. 3. True course, North; variation, iS" CMstrrly. CouusE DY Compass. N. iS° \V. Ex. 4. True course, West ; variatiou, 2.° westerly. Course by Compass, N. 6y° Vf, Deviation nf the Compass, 222. From the earliest times it was known that Jf a magnet, or a piece of ordinary iron, were brought near to a compass, it would deflect the needle in its pointing, and so make the compass iiidiratiims en'oneous. Compasses on board ship, therefore, wcu-e IK it jilaccd near to each other, and iron was rigorously kept away from their vicinity. With these precautions, though accidents Bometimes happened from iron in the vicinity of the couipass being overlooked, ships were navigated with a fair amount of security. But as iron became increasingly used in the construc- tion of ships, and by the introduction therein of steam engines, with their boilers and funnels, it was no longer possible to navi- gate, without systematically allowing for the deflection of tli« compass needle caused thereby. The horizontal angle, which the needle is deflected by the iron in or of the ship, is called the Deviation of the Compass. It is named easterly or positive (E. or +), when the north end of needle is deflected to the eastward ; and westerly or negative (W. or -), when deflected to the westward. The mode of ascertaining ami applying the deviation of +he compass, is the next problem to engage the alte ilion of the student of navi-atn^n. 74 NAVIGATION. WitLin half a century of the present time, many navigators dDubted the existence of the deviation of the compass; or, while admitting its existence, denied that it was of any practical importance. And the belief was not uncommon, that it was a constant error- -that is, that it was the same in amount with the ship's head in any direction. Those, however, who had studied tlie subject, or whom circumstances had made familiar therewith, atknowledged its importance, and recognised the necessity of iiscertainiiig the deviation of the compass, with the ship's head in all directions. 223. There are three standards from which to reckon an angle of direction. First, from the meridian, the direction of which can always be ascertained astronomically. A course or bearing tlius reckoned, is called a true course, or true bearing. Second, irom the direction of the magnetic north ; that is, from the direction of a magnetic needle, when uninfluenced by any con- tiguous iron, or by any such local disturbances as are mentioned in No. 220. A course or bearing thus reckoned, is called a nisignetic course or bearing. Third, from the direction of the compass needle, as shown by a compass which is instruinen- tiilly correct, placed in any position. A course or bearing thus rocK'oned, is called a compass course or bearing. The prefix correct may be placed to either of these quan- tities. The terms correct true, correct magnetic, correct com- piss, are used to distinguish the exact angles from those more or less approximate. The student must not confuse correct compass with magnetic. A correct compass course or bearing means a course or bearing accurately observed, with an accurate compass, regardless of any disturbance by which the compass may be influenced. 224. From the fact that compasses, in different parts of a ship, gave different indications, came the necessity for navigating by one especial compass, placed in a selected position. Such a compass is called the Standard Compass. It should be an azimuth compass, that is, one fitted for observing bearings ; and one essen- tial of its position is, that from it bearings can be taken all round the horizon, and at any altitude. Turning a ship round, so as to place her head on all points of the compass in succession, for the purpose of ascer- taining the deviation, is called swinging the ship. A ship may he warped or towed round, when lying at anchor or at moor- ings ; or advantage may be taken of her turning with the tide. Wherever there is room, it may be convenient to steer a ship n)und under steam. It is in all cases desirable that the slii[) should be checked in her swinging, and steadied on the point ou which it is desired to obtain the deviation. As the variation of the compass is determined by coni- [laruig the true bearing of an object with its uiiignetic bearing, IKSTKUMEXTS OF NAVIGATION. I 'J SO Ibe dori.dion of tlie compass is ascertained by compnrinnf the niaLTiiPtic bearing witb the compass bearinj^ — the compass, at Ibe time, beini^ deflected by the iron in and of the ship only. Any other disturbance, such as from the proximity of other ships or masses of iron, or the irregular influence of the land, is not devialion according to the definition already given. The first problem is, therefore, to determine the magnetic bearing of some object external to the ship. The sun is very commonly used; the true bearing is easily found, and the va,ria- tion being applied thereto, gives its magnetic bearing. A dis- tant mark on the land may also be used ; its true bearing may be found by the chart, or by measuring and applying the liori- zontal angle or difference of bearing between it and the sun, and the magnetic bearing by further applying the variation. A third method is to have a correct compass in a convenient posi- tion on shore, v^here it is free from magnetic disturbances. Then the bearing of that compass being taken from the standard compass, and the bearing of the standard compass being simul- taneously taken from the shore compass, the deviation of the standard compass is found by comparison. These methods are spoken of as, swinging by the sun, swinging by distant mark, and swinging by shore compass. When using a distant mark, it should be so far away that the radius of the circle, along the circumference of which the standard compass moves as the ship goes round, subtends a smaller angle than is of practical consequence in navigating. Otherwise the bearings must be corrected for parallax. There are many places where the true direction of lines, on which two known and conspicuous marks appear in one, are known. These lines, called transit lines, offer especial facilities for ascertaining the deviation. Looking from the centre of the card, if the bearing shown by the compass is to the left of the magnetic bearing, the needle is obviously deflected to the right, and the deviation con- Bequently called easterly. If the bearing shown by the compass IS to the right of the magnetic bearing, the needle must be deflected to the left, and the deviation westerly. 225. Though the deviation of other compasses is not of so much importance as that of the standard, it is usual to note the direction of the ship's head, as shown by them, when it is on each point by the standard. The deviation is usually tabulated for reference, in some form similar to the following, which ia com- monly called a Deviation Table. nraa hy St;.,„l.>r.i Cuul|..lM DcTiat Cum pZl Dirccllo . of Head by other Compasses 1 r..rt Steering L larboard Steering Bridge Ciiii. 1 1 76 UAVir.ATTON. The bearing-plate is frequently used in swinging. The vanea on the bearing-jihite, beinc- set to the known magnetic bear- ing of the sun, distant marJi, or shore compass, the magnetic direction of head is sliown by the lubber mark, when the plate ifl turned round so that the vanes point to the object. Tlius, Ihe deviation of the compasses on the magnetic points is sliown, and may be tabulated as follows : — Head Magnetic Direotiou of Head by Coinpisaes St,u dard Port Steering Starboard Steering Bridge Comp.n^ 1 226. It is customary to form a deviation table from observa- tions made on each point. But it may be convenient, or neces- sary, to form such a table with fewer observations, such as on every second or third point. Further, it may not be possible to get the observations exactly on the points. The problem, there- ft ire, is to form a deviation table with few observations, irregularly- distributed round the compass. This is done by drawing a curve of deviations in the following manner. Draw a vertical line on paper, and divide it as a com- pass card is divided. The vertical line will thus represent the circumference of the card unrolled, and formed into a straiglit iiue. Through each compass point draw a line at right angles to the vertical line. On these lines, with any convenient scale, lay off the deviation found on each point. On parallel lines, passing thi-ough any intermediate degree or division of the point, lay otf the deviation found thereon. Easterly deviation to be measured from the vertical line to the right, and westerly deviation to the left, marking, by a cross or otherwise, the positions thus deter- mined. Now draw a line which, without being irregular in direction, passes most nearly through the several marks. This line, in practice, will always be a curve. The distance of the point of intersection of this curve with any point line, from the vertical line, will give the deviation on that point, using the same scale as before. Example. — The following deviations having been observed, tiud the deviation on eacli compass point. North S O E N ■■{ E 6 o E NNE 10 o K NE 1 N •4 o E ENK 12 E E^N lO o E EbS 9 so E SEbE 6 E SE I o W «SE 2 oo W ■South o W SW 1, 3 o w SW ■,. \v AVSW o w We.t lO o w W bN- N II o \v NW 10 o \v NNW i W 6 3t> w ~ " " " ^°^^'"'. ■;;7";;\r ""^ ^\ X. — - X 1 T 1 '/ .__ ^^^^ fg^f / / / / y^ 1 : South icule i}/J)ena&/^ri, - 1 J / J ■ / '■ 1 ■. f : — r J V '■ X- — : ^^^ : -North A ::rr-. NAVIOATION. 227. Plotting these observations in the mannei" directed, and as shown in the foregoing diagram, the following table of devialioiis is obtained. North 5 o E NbE 7 45 E NNE lO o E NEb N 12 15 E NE 13 30 E NE h E >3 30 E ENE 12 45 E EbN II 45 E East 10 30 E EbS 845 E Eb E 7 15 E SEb E 4 30 E SE 45 E SEb 3 2 W SSE 3 30 W S b E 4 30 W South 5 S b w S ssw 5 S\V b 8 5 SW 5 30 S\V b w 6 30 W8VV 7 45 W bS S45 West 9 45 W bN 10 30 wxvv 10 45 NW b W 10 30 NW 9 45 NW b N 8 NNW 5 '5 N b VV 1 In the diao-ram shown, the vertical scale is made small as com- pared with the horizontal scale, in order to get it within the limits of the page. A sheet of ordinary ruled foolscap will be found very convenient for plotting deviations to form the curve. 228. The methods of ascertaining the deviation having been explained, the following are directions for applying the same to a compass course or bearing, so a? to obtain the magnetic course or bearing. The ship's head being on any compass point, and the devia- tion on that point being easterly, that deviation must be allowed to the right, to find tie magnetic direction of the ship's head ; and also to the right of any bearing taken by compass, to find the magnetic bearing. If tlie deviation on the compass course is westerly, it must be allowed to the left, to find the magnetic course or bearing. Example. — Shiji's head E.N.E. by compass, a point of land bore N. 10° W. What is the magnetic direction of the ship's head, and the magnetic bearing of the point, the deviation being as given in table 227 ? The deviation on E.N.E. is 12.45 E., which allowed to the right of N. 67.30 E., gives N. 80.15 E. as the magnetic direction of the ship's head; and allowed to the right of N, 10.0 W., gives N. 2.45 E. as the magnetic bearing of the point. In the same way, head being N.W. and bearing S. 40 E., the deviation on N.W. is 9.45 W., which allowed to the left, gives N. 54.45 VV. as magnetic direction of ship's head, and S. 49.45 E. as magnetic Lieanng of point. To turn mugm-tic course's or bearings into compass couisca or rN'STlUlMENTS OF NAVIGATION. E3 courses or bearinsrs into b*-arings, it is obvious that the deviation shoulil be allowed tbo opposite way. That is, easterly deviation to the left, and westerly deviation to the right. 229. To facilitate turning conij magnetic courses or bearings, and the reverse, certain graphic methods are sometimes used. The most com- mon is one called, from its inventor, Na|)ier's diagram. The example given, wherein are plotted, through a quadrant, the observations given in No. 22t), shows the use of this diagram for the purjiose named, as well as for forming a curve of devia- tions from few observations. The dotted compass point lines intersect the vertical line, at an angle of 60°, and the vertical scale and deviation scale areequal. There- fore, if the deviation found on any compass point be laid off on one of the dotted lines, or on a line paral- lel thereto, and, from the point reached, a line be drawn making an angle of 60° with the compass jioint line, it will intersect the ver- tical line at the magnetic point. ^- And, vice versa, if the deviation on -^ a magnetic point be laid oS" on one of the plain lines, or on a line parallel thereto, the return line, drawn as '" before, will reach the vertical line '' Rt the compass point. The three lines form an equil.ateral triangle, ..^ of which the ditference between compass and magnetic forms the base, the other sides being equal thereto, and to the deviation due -^ to the direction of head, whether ^' given by compass or magnetic. 2:30. Another method, called the straight line method, is due to Mr. Archibald Smith. Tt is only useful for showing, at a glance, the mag- netic course equivalent to any given compass course, and vice versa, when the deviation is known. Tt consists merely of two parallel vertical lines, each divided as ti.o circumference of a compass card is divided. Strai-ht lines ute 80 KAVIGATION, STRAIGHT LINE DIAGRAM. COMPASS AfAGN£T/C XOURS£S , */Cb N drawn, from the compass points on one Hue to the ma^-netic points on the other. In the annexed example, the deviation table through one quad- rant, given in No. 227, is thus treated. If a ship be steering any- compass course, shown on the lett-liand column, the corre- sponding magnetic course is shown on the right-hand column. NOfiT/j ^"'l if i*' is desired to steer any- magnetic course, shown on the right-hand column, the required conniass course is shown on the left-hand column. 2ol. A third method is to have two prints of compass cai-ds, one laid on the other. The upper card somewhat smaller than the lower, and capable of being rotated about the commdu centre. The lower card, being fixed, may be considered as re- presenting either true, or mag- netic, courses or bearings. Consider the lower card to represent true courses and bear- ings, and the north points of the two cards togethei-. Conceive ■/v£i£ the north point of the upper card, moved thiough an arc equal to the variation, away from the north point on the lower card, to the right when the variation is easterly, and to the left when the variation is westerly. Magne- tic courses and bearings on the upper card, and true courses and bearings on the lower card, will now be coincident. Similarly, if the lower card be considered as showing mag- netic courses or bearings, and the north points of the cards be separated by an arc equal to the deviation, then the compass courses and bearings on the upper card, will coincide with magnetic courses or bearings on the lower card. V£{>£ /V£6N f/£ £6M £AST rNSTnUMKN'TS OF NAVIOATIO^T. 81 DiMi^TMms on wliicli cnrvcs of deviation can be flr:iwn, so ris to show iiiilitf'tTOnt obsorviitions, and tiiiis elitiiiiuite their elU'cts, or to form the curve from few observations, are of un- doubted value to the seaman. But it is a question, whether any means such as have been described, for turning ma<Tnetic courses into compass courses, or the reverse, are of ultimate benefit. Tlie habit of considering the effect on courses and bearings, of the north point, and consequently the whole circumference of the card, being turned right or left, from what may be c<.ti- sidered its proper position, so as to have a clear conception thereof in the mind, will make the seaman independent of rules, ftud of all such semi-mechanical methods. Adjustment of the Compagf.. 232. If the increase of iron put into ships bad been limited to engines and boilers, it is possible that a compass might have been Bo placed, in most ships, that the deviation would have been com- paratively small. Seamen might have continued to navigate with confidence, by ascertaining and applying the deviation. But when ships were built with iron beams, iron frames, or wholly of iron, it was no longer possible to evade a deviation so large as to be unmanageable ; and steps had to be taken to correct, or, as it is now called, adjust, the compass. This operation is generally performed by practised compass adjusters ; but many rightly think this is essentially the duty of a seaman, and that he should also have sutficient knowledge of magnetism to enable him to select the best position for the com- passes of a ship. In a book in which teaching navigation is the main object, magnetism can only be treated with brevity ; but it is hoped that the navigator will find herein all that is required for his guidance. The horizontal pointing of the compass needle has been shown to be of the utmost importance to the navigator. For the right understanding of the magnetism of iron ships, however, and its efl'ect on the compass, some further knowledge of the pointing of the magnetised needle, and the cause thereof, is necessary. In the year 157G, Robert Norman, a mathematical instrument maker, of London, discovered that a needle, however nicely balanced, would, after being magnetised, depart from the hori- zontal, and assume a position within 20° of vertical. By careful observations he found that the needle in London, at that date, pointed, with its north end downward, 71°"50' from the horizontal. Since that time, observations have been made neai'ly all over the world. It is found that the needle is horizontal only on a line round the earth, not far from the equator. Going from this line to the northward, the needle points with its north end downwards ; and going to the southward, with its south end downwards. The angle of inclination, in both cases, increases, till in a position in 82 NAVIGATION. each hemisphere, about 18° from the earth's poles, the needle becomes vertical. These positions are called Magnetic Poles. This angle of inclination to the horizontal is called the Dip. It is named positive, or +, when the end towards the north magnetic pole is the lower, and negative, or — , when the end towards the south magnetic pole is the lowei-. Lilt e the variation, the dip is found to change with time, and other circumstances. In the adjacent maps, lines of equal dip are drawn. The line whereon the dip = 0, is called the magnetic equator; and the lines of equal dip may be considered as parallels of magnetio latitude. The lines running nearly north and south show the horizontal direction the needle lies in, and may be considered as magnetic meridians. These lines converge to the magnetic pole in each hemisphere. For the use of seamen, there is no better way of giving the variation of the compass, than by lines of equal variation, as drawn on the variation chart (No. 2-38) ; but the lines here shown give a more direct representation of the pointing of the compass needle. 233. In the beginning of the present century it became known, chiefly through the researches of Humboldt, that the strength, or force, with which the needle points is not the same in all parts of the earth. It may be stated, generally, that this force is least about the equator, and, like the dip, inci-eases towards the poles. Also, like the variation and dip, it is not constant in value at the same place. The line whereon the magnetic force is least, coincides nearly with the magnetic equator ; but there are apparently, in each hemisphere, two points whei-e the force is greater than in the sur- rounding regions, neither of which coincides with the magnetic pole. As the earth's force is not horizontal, except at the magnetic equator, it is convenient to reckon, or resolve, as it is called, that force in the horizontal and vertical directions. If the length of the line AC represents the earth's force, and the angle A be equal to the dip, then the horizontal line A B, and the vertical line B C, will in length repre- sent, respectively, the horizontal and ver- tical components of the earth's force. These quantities are usually called the Horizontal Force, the Vertical Force, and the Total Force. Of these quantities antl the dip, if any two are known, the other two may be found by the ordinary pro. cesses of trigonometry. As previously stated, the dip and total force increase, going away from the magnetic equator ; but it is evident that when the dip is 90° the horizontal force must vanish, whatever the total force may Hemisphere from 60° W. to 120° E. Longitude. -/f? <fA i f i ^ ^^A^ 7n<: ■ ■ J \3\ ^ V^V^ X -^^ )y/J 1 1 \ \ ^ V^s^^ \ \ \ ""/r^^^^^^ h /^-i W 4mrTu^ W>^° /^ ■i r fa i 4 ^^ ^ T LJ w- fe^ 'Tj] — -S| %J'\^ — i^ PT y ^ ^ ^rt- tf^ , r \\ 20 l-Jr^4f^ ^xK vV -m \ \/> <w — tv~JL / iKc / J / / //Wsf/ \V^ -v\^ A^i /f\iy :>^ \v' K^ \\jy(^TTX ffO^^ ^ ^ ^ u ^ \M^ Maps showing the Magnetic Equator, lines ot Equal Dip, and Horizon- tal Direction of the Compass Needle. The parallels of latitude and the meridians are drawn at every fifteen degrees of latitude and longitude; the figures at the circumference denote the dip in degrees along the respective magnetic parallels ; and the direction of the magnetic meridians, compared with the direction of the gepgraphical meridians, shows the variation. Hemisphere from 120° E. to 60° W. Longitude. The points (©) to which the magnetic meridians converge are the magnetic poles, sometimes called, from the dip thereat being 90°, the poles of Verticity. The points ( » ) show the approximate position of the foci of maximum force. It is remarkable that these ^x points are within 160° of longitude. These maps, and the following table of horizontal force, are based on the good work on this sobject done by the late Sir K. Kvans, R.N. INSTKUMKNTS OF NAVIGATION. S3 be. The dip and total force, therefore, increase together in such a manner that the horizontal force continually diminishes. Tlie horizontal force is the only part of the earth's force by which the compass card maintains its due position. The sea- man is generally satisfied if this condition is fairly answered; but he must be sometimes painfully aware, from what is called the sluggishness of his compass, that this force is, at best, very feeble. The following table gives the comparative value of the hori- zontal force, in different positions ; the maximum value being considered as unity. COMi'AKATIVE VALUE OF HUKIZONTAL FORCE Ma ximum Value equal Unity East Lonu TUDE 1 Latitude 0° 040 041 3U° 0-43 •15° 60° 044 75° 043 90° 042 105° 043 120° 044 135° 0-4S 150° 047 165° 049 180° 048 N 60° 044 50 049 052 055 0-57 058 058 058 057 057 059 060 0'6q 0-59 40 060 064 0-6S 070 073 075 075 075 074 073 071 070 06S 30 071 075 079 083 087 089 090 0S9 087 084 82 oSo 076 20 07S oSi 0S6 90 094 097 097 096 094 092 0-89 086 0-84 10 0S2 085 o-SS 090 093 097 I 00 I 00 099 097 094 092 090 079 079 080 083 0-87 090 096 roo I 00 I 00 0-98 0-97 0-95 S 10 070 070 070 072 076 082 088 092 096 0-98 0-98 097 096 20 06 1 0-59 0-59 60 0-64 071 077 082 087 090 091 091 090 30 056 054 052 0-53 054 058 0-52 06S 071 075 078 oSi 0S2 40 0-5S 051 0-49 04S 049 0-50 051 054 055 057 062 065 008 SO 054 050 047 0-44 043 042 041 039 038 039 0-43 047 052 60 OS3 0-49 046 042 0-39 036 032 0-27 0-20 024 030 037 West Long ITUDE Latitude 1 0° 15° 0-37 30° 033 45° 0-29 60° _75^ 012 90° 010 105° 020 120° 029 135° 0-38 150° 165° ISO" 0-48 N 60° 040 0-43 046 50 049 0-45 041 037 034 032 036 043 049 0-54 055 057 059 40 060 055 050 0-48 047 050 056 060 064 065 065 obO 0(8 30 071 066 062 061 065 071 077 080 080 077 074 074 076 20 07S 074 072 072 077 085 090 090 08S 085 082 0S2 084 10 S2 079 077 079 084 089 094 0-95 092 0S9 0S9 090 90 79 07S 077 o-8o 083 089 0-92 093 092 090 092 093 09i S 10 70 071 073 076 080 084 0S7 088 089 88 091 093 096 20 61 065 0-68 070 074 079 082 0-84 08s 087 088 ogo 090 30 $(' 060 063 067 070 076 077 07S 07S 080 081 0S2 082 40 55 059 063 067 071 074 075 073 071 070 070 070 068 50 54 059 0-63 o6g 072 073 0&8 0O3 061 0-59 056 0'52 60 053 0-5S 063 0-6S 07. °-' 067 060 055 050 040 042 o'37 234. In dealing with the subject of compass adjustment, it will sometimes be useful for the seaman to know the value of the force with which the needle points on board ship, compared with the force with which it points on shore; or the force vvith which 84 NAVIGATION. it points when the ship's head is in one direction, compared wilh the force wit)i wliich it points when the head is in other direc- tions. It is necessary, therefore, to show how comparative mag- netic force is measured. If a magnetised needle, balanced on ila centre, be disturbed from its position of rest, it will, like a pendulum, vibrate through diminishing arcs, till it again comes to rest. The speed of the needle is increased when the magnetic ibrce is increased ; the force being proportional to the squax-e of the speed of the needle. That is, if the needle in one position makes 10 vibrations in any given time, and in another position makes ] 2 vibrations in the same time, the magnetic force in the first position is to the magnetic force in the second position as 10^ is to 12». It is convenient to measure the horizontal force and the vertical force separately. The horizontal force is measured by means of a flat and pointed needle, about three inches long. It ban a jewelled cap at its centre, which works on a sharp pivot. It must be used in a covered box, or compass bowl, to protect it from the motion of the air. It is brought horizontal by a small weight, counterbalancing the dip, and so vibrated in the horizontal plane. Horizontal force may also be measured by deflection. If a magnet be placed at right angles to the direction of the needle, the magnet will deflect the needle through a certain angle, depending upon the strength of the magnet, compared with the horizontal force. The smaller the force, the larger the angle of deflection of the needle, the force being as the cosine of the angle of deflection. Or the deflecting magnet may be moved round, and kept at right angles to the compass needle, and the horizontnl force measured by the maximum deflection the magnet is capable of producing, when thus applied. Vertical force is measured by means of a Dip Circle. This is an instrument having a flat pointed needle, with an axle passing through its centre of gravity, about which it can rotate in a vertical plane ; the axle being supported at the centre of a graduated circle. If the circle is placed in the vertical plane of the magnetic force, the needle will stand in the direction of that force, showing the dip, if it be acted on by the earth's force only. A small weight placed on the upper arm of the needle, bringing it horizontal, will be a measure of the vertical force. If the circle is placed at right angles to the plane of the magnetic force, the needle will hang vertically, where there is any vertical foice, and in this position may be vibrated, so as to measui-e that force. Measuring either horizontal or vertical force by vibration, the initial arc should be the same, in any positions wherein it is desired to compare those forces. The effects of friction, and the resistance of the air, are to cause the needle to take a little more INSTHUMENTS OF NAVIGATION. 85 time, in going throtigh tho larger arcs than the smaller, and altiniately to bring it to rest. The smallest arcs which can be conveniently used give the best results. 235. Studying the phenomena of the pointing of the magne- tised needle on the earth's surface, and comparing them with the effects of one magnetised needle, or steel bar, on another magne- tised needle, or steel bar, the conviction gradually giiined ground, that the earth is, or has the properties of, a large magnet. Those properties are two. First, Attraction and Repulsion : the property by which one magnet will attract and repel another, according to definite laws. Second, Induction : the property by which a magnet can impart magnetism, and so convert into a magnet liny piece of iron or steel, either by contact or mere proximity. The property of attraction and repulsion may be shown, by bringing two compass cards near to each other. The north part of one card will push away or repel the north part, and attract or draw towards it the south part, of the other. The ends of magnets are called poles, and we express the law of attraction and repulsion by saying, like poles repel, and unlike poles attract, each other. This attraction and repulsion may be due to two different kinds of magnetism in the poles, or to an excess of magnetism in one pole as compared with the other, or it may be a magnetic state, depending upon neither one cause nor the othex-. It will be convenient to spjak of the magnetic state of the north pole of the compass needle as positive, indicating it by the ti'gn +, and of that of the south pole as negative, indicating it by the sign — . The pointing of the magnetised needle appears to be, the direction it takes up in obedience to the law of attraction and repulsion existing between it and the larger magnet, the earth. Also, the increasing strength, with which the needle is found to point as the latitude increases, appears due to the approach to the magnetic poles of the earth. By the law of induction, a magnet when brought near to any piece of unmagnetised iron, induces magnetism therein; the ne<ar pole of the magnet, and the proximate part of the iron, having magnetism of opi>osite kinds. The similar magnetism to that of the near pole of the magnet is found in a remote part of the iron. Applying this law to the earth as a large magnet, the magnetism of iron and iron structures is apparently due to in- duction from the earth, and the end or part of ii'on which is towards the north will have positive magnetism. In dealing with the magnetism of iron ships, this property of induction, hitherto little thought about by seamen, becomes of great importance. The earth's magnetic force, by inducing inagiietism in the iron uf a ship, is the source of all magnetic disturbances of the compass. 86 KAVIGATION. 236. The question as to how the earth became magnetisecl will perhaps come into the mind — possibly it is, or was, magnetised bj induction, from some far distant cause. But magnetism may be induced by electricity. If an insulated wire is passed round a piece of iron, and the wire be considered as conveying an electric current flowing from positive to negative, the iron will become magnetised, and have positive and negative powei'S, as shown in the figure. If tlie trade winds flowing round the earth from the eastward, be considered as acting as a positive electric current, the earth would be magnetised with a negative pole to the north, and a positive pole to the south. Whether it is thus magnetised or not, the idea will aid the memory as to the magnetic state of the earth, show how magnetic forces may be generated by electricity, and suggest the possibility of compass disturbance, by the in- creasing use of electricity on board ship. 237. All iron is capable of receiving magnetism by induction from the earth. If the iron remain a long time in the same posi- tion, or if it be hammered or subjected to mechanical violence, part of the induced magnetism will remain. That is, the iron will show polarity in the same parts, after it has been moved into another position, relatively to the line of the earth's force. All magnetism, therefore, may be called induced magnetism. That which instantly passes awaj^, when the inducing cause no longer acts, is called transient magnetism. That which remains for a longer or shorter time, is generally called permanent mag- netism. The tevm permanent, in this extended sense, means all mugnetism that is not transient. The terms trans-permanent, sub-permanent, and permanent, may be used to indicate increas- ing degrees of permanency, if desired. It is, however, a question whether anything is gained by thus multiplying terms, as no definite line of separation can exist. Speaking generally, iron will receive or part with magnetism more or less readily, according as it is soft or hard. Hard iron or steel, when magnetised, will retain its polarity for a very long time. 238. The disturbing effects of iron on a compass, being caus(id by magnetism induced in the iron by the earth's magnetism, the possibility of so placing iron about a compass on board ship as to counteract the effect of the iron of the ship, is the problem of compass adjustment. Professor Barlow was the first to deal practically with compass adjustment, and the problem was subsequently completely solved by Professor Airy in 1839. That gentleman gave the results of Ilia researches and experiments in the following words : ' Bj INSTUUMKNTS OF NAVIGATION. 87 placing a magnet so that its action will take place in a direction opposite to that which the investigations show to be tlie direction ot the ship's independent magnetic action, and at snch a distance that its etiect is equal to that of the ship's independent magne- tism, and bj counteracting the efl'ect of the induced magnetism by means of the induced magnetism of another mass [according to rules which are given], the compass may be made to point exactly as if it were free from disturbance.' Briefly, this state- ment is to the eflect, that the permanent magnetism of the ship may be counteracted by the permanent magnetism of steel mag- nets, and the transient magnetism of the ship by the transient magnetism of iron ; the magnets and ii-on being placed near the compass, according to definite rules. In order to be able to consider together, the disturbing effects of the iron of the ship on the compass, and the action of magnets and iron in counteracting the same, a brief explanation of the latter is necessary, 239. Magnets, when used to adjust a compass, are applied, generally, either end on, or, as it has been termed, broadside on. If a magnet be placed near a compass, so that the centre of the needle is in the line of the magnet, the etfect of the magnet is to cause a force pushing away the north point of the needle, if the positive end of the magnet is presented, and drawing the north point of the needle towards the magnet, if the negative end is presented. In the figure, | if c represent the centre ^-^ — i-" ^ ^ of the compass needle, the arrows represent the di- C rection of the force on its ^ north end. This is called the end-on position of the magnet. If a magnet be placed near a compass, so that the centre of the needle is in the same plane as the magnet, and on a line drawn from the middle of the magnet, perpendicular to its direc- tion, the efl'ect of the magnet is to cause a force parallel to itself, pushing ■< ^ • the north end of the needle away from i the positive end of the magnet. In the i figure, if c be the centre of the coin- : piiss needle, the arrow shows the di- ; rection of the force on its north end. i This is sometimes called the broadside [ - ' +"1 position of the magnet. Magnets us.'d for compass adjustment are made of hard steel, and well ningnetised. Their magnetism may be considered as per- manent. Thus, by means of a magnet, a permanent magnetic force can be produced, pushing the north end of the compass needle in any desired direction. 240. The iron used in adjusting compasses should be soft malle- 88 NAVIGATION. E. mag. laU -f" . msg. Ui. 4- able iron, so that magnetism is readily induced therein by the carth^s force, and readily parted with ; that is, it does not become permanent. It is used for two jjurposes. For one purpose, it is in the form of an upright bar, placed, generally, before or abaft the compass. For another purpose, masses of chain or scrap iron in boxes, cylinders, or spheres, are used. These are placed besidt the compass, on the same level as the needle. 241. The action of the upright bar depends upon the earth's rei-tical force. In north magnetic latitude, the lower end has positive magnetism, and the upjier end negative magnetism. On the magnetic equator the bar may be considered as unmagnetised. In south magnetic latitude, the lower end has negative magnetism, and the upper end positive magnet- ism. Therefoi-e a magnetic force in any direction cau be produced, acting on the north end of the compass needle, varying in strength with the earth's vertical force, by placing the upper end of the bar in a suitable position. It is generally desired to make this force horizontal, as shown in the figure, where c is the centre of the compass needle. After Captain Flinders, R.N., who was the first to propose this, or, indeed, any mode of counteracting the effect of the ship's iron on the compass, iron thus used is called a Flinders bar. 2-12. The action of iron placed beside a compass, is not quite so simple as that of the Flinders bar. In the fig. let c be the centre of the compass needle, and the circle the outer circumference of the binnacle. Let a i-epresent a horizontal iron rod, placed radially north of the centre of the compass. In this position it will be magnetised by in- duction from the earth — the north end of the rod with I^ositive magnetism, and the south end with negative mag- netism. It will cause no deflection of the needle, be- cause the force is in the line of the needle. It will, how- ever, increase the force with which the needle points. :n: INSTHUMKNTS OF NAVIGATION. Conceiving the rod to be moved round the needle to the rij;bt, as the spokes of a wheel move round its centre, it will be seen that the amount of n)a<^netism in the rod will dimhiish as it goes round, till in the east position it may be considered as without magnetism. But as the rod leaves the north position, so the magnetic force of the rod, by being inclined to the needle at a greater angle, has a greater proportional efl'ect in deflecting it. From the combined action of these two causes, the maximum deflection of the needle occurs when the rod is in the N.E. position. Following the rod round, and noting the magnetism induced thereui by the earth's magnetism, and the etfect of the magnetic force, thus generated in the rod, in detlectiiig the needle, the fidlowing results will appear: — Rod North or South of the centre of the needle. Increase of force, no deflection of the needle. Rod N.E. or S.W. Increase of force, maximum easterly deflection of the needle. Rod East or West. No etfect on the needle. Rod S.E. or N.W. Increase of force, maximum westerly deflection of the needle. Thus it will be seen, that the effect of the rod is to cause a deflection of the needle, easterly and westerly in alternate quad- rants, and to increase the mean magnetic force. It will also be Been, that the etfect of two rods opposite to each other, is to double the effect of one. 243. Another instructive example of the effects of iron moving rod ph, .•ed tangentially. N round a compass is that of a similar Following the rod round, and noting the magnetism induced therein, and the effect thereof on the compass needle, as in the figure, the following results will be seen : — Near end of the rod North or South of the centre of the needle. No effect. Near end of the rod N.E. or S.W. of the centre of the needle. Westerly deflection of the needle. Near end of the rod East or West of the centre of the needle. Maximum westerly deflection of the needle. c. Near end of the rod S.E. or N. W. '^ of the centre of the needle. Westerly deflection of the needle. In this example, as in 2 12, the effect of two rods in opposite positions is to double the effect of one. When two rods thus placed tangentially, having their near 90 NAVIGATION. ends at North and East, or in any positions 90° apart, revolvG about the compass together, one rod will cause a maximum deflection of the needle, when the other rod has no effect thereon. As the effect of one rod increases, the effect of the other decreases ; and the combined effect of the two rods, thus revolving together, is a constant westerly deflection of the needle. If the rods are placed in the opposite direction from their point of contact with the circle, similar easterly deflections will be produced. The ii-on rod h.is been here used as an example, be- cause the e9"ects can be most simply shown thereby ; but other forms of iron (240) are generally used in adjusting compasses, to produce the same effects. Hollow iron spheres are used with Sir William Thomson's compass. Their action is that of a rod of the length of the diameter of the sphere, always standing in the line of the earth's magnetic force, and magnetised thereby. In the figure, where c is the centre of the compass needle, it will be seen that the hori- zontal force of the spheres deflects it, and aflTects its pointing force, in the same manner as the iron rod in 242. When east and west of the centre, however, sjiheres diminish the directive force on the needle, more than the forms of iron commonly used. Having briefly examined the means employed to counteract the ship's magnetic forces, the origin and effect of those forces, and the mode of applying the counteracting means, may now be considered. 244. An iron ship, in the course of construction, stands in the influence, or field as it is termed, of the earth's magnetism, and is consequently magnetised by induction. In north magnetic latitude, all upright iron structures, such as stern-post and frames, have positive magnetism in their loiver ends, and negative mag- netism in their upper ends. In south magnetic latitude, these conditions are reversed. In all latitudes, horizontal iron structures, such as beams and keel, have positive magnetism in theirnorthern ends, and negative magnetism in thinr southern ends. The ship throughout is, in course of building, permeated with magnetism in the direction of the inducing force. Part of the magnetism thus acquired in building remains after the ship has been launched, causing a permanent magnetic force, in some direction in the ship. INSTRUMENTS OF NAVIGATION. 91 lliis force tends to draw the north point of the compass towards that part of the ship which was south in building. Besides this permanent magnetism, the ship, as she sub- sequently turns about with her head in different directions, takes up magnetism according to her varying positions. The amount of magnetism iron will thus receive by induction, within the limits of the change in the earth's force, varies as that force ; the ends of beams, and other parts of the ship's structure, which are towards the north having positive magnetism, which changes and becomes negative when the direction of the shiji's head is reversed. It ia evident, however, that vertical iron will have magnetism which does not dejjend on the direction of the ship's head, bi t which will vary, in character and value, with the earth's vertical foi-ce only. 245. From these premises it will be seen, that there must be always a Constant force, and a Variable force, acting on the compass needle as the ship goes round. Therefore, if the direction and value of these forces are known, together with the law \rhich governs the change in the variable force, the deviation of the compass could be found without swinging the ship. Generally, it is easier to deal with the deviation than with the forces which cause it ; but a knowledge of the manner in which these forces act, facilitates very much the construction of a deviation table. Considering the commercial value of time, in all matters relating 10 shipping, this is a subject of no small importance. 24(5. It has been stated, that j^art of the magnetism acquired in building causes a constant force, in some direction, in the ship. The amount of deviation any force is capable of producing must decrease, as the force with which the needle points increases. Therefore, the deviation caused by the ship's permanent magnetism varies inversely as the earth's horizontal foi-ce. It is also clear, that, if the direction of the ship's permanent magnetic force is known, a permanent force by means of magnets (239) might be produced to counteract it; and if this magnetism of the ship, and of the magnets, were equally permanent, the adjustment would be perfect for all time and places. The transient magnetism of vertical iron also causes a force which is constant in direction and value, as the ship goes round. This force, however, changes with change of place, as it depends on the earth's vertical force for its value. The liability of the needle to be deQected thereby varies inversely as the horizontal force. Therefore, the deviation caused by the tran- sient magnetism of vertical iron will vary as, ver. force x :; =tan. dip hor. torce 247. The following diagram will show how the compass is ^^('.■^('d Ijy the transient magnetism of vertical iron, and the niiiiinii- ill which Flinders' bar (2 11) ctuiutcracts that cll'cot. 32 NAVIGATION'. Afteb Part of Ship's Upper Deck. Head East. North magnetic latitude. e--k- South magnetic latitude -(±>- s In north magnetic latitude, the upper partcf the ship's frames having negative magnet isui, a compass in the position (a) would liave its north point drawn to the westvrard. In south magnetic- latitude, it would be drawn to the eastward. It is certain that no fixed magnet would meet this change. A Flinders bar, how- ever, might be placed before the compass, so that its magnetism would exactly counteract that of the stern frames. The mag- jietisin of the bar would change, exactly as that of the stern frames, when the ship went into south magnetic magnitude. At a position (6), generally rather more than one-third of the distance between the stern and the funnel (/), the magnetism of the upper part of the boilers and funnel counteracts that of the stern frames, so that no bar is required. At the position (c), tlie bar would be lequired abaft the compass; at the position (<Z), before the compass. The position (fc), when not otherwise objectionable, is chosen for the position of the standard compass in the Royal Navy. The position (t/), being more convenient, is commonly used in the Mercantile Navy. If a compass were placed out of the middle line, its north point would be drawn to the near side of the ship in north mag- netic latitude, and repelled therefrom in south magnetic latitude. This effect would have its maximum value when the ship's head is north or south; and the Flinders bar must be towards the middle line, to counteract it. 248. The horizontal forces, from permanent magnetism and the transient magnetism of vertical iron, cause a deviation which is zero when the direction of the ship's head is such that the resultant of these forces is in the north and south line ; and a maximum deviation when that resultant is in the east and west line. This deviation, from being easterly through one semi- circle, and westerly through the other, is called the Semicircular Deviation. In correcting the semicircular deviation, such magnets as are commonly used should not be brought nearer than twice their length to the compass needles. And Flinders' bar should not be 80 near to the compass needles, or correcting magnets, as to receive m;.,gnetism by induction from them. 210. When the semicircular deviation h;i,s been got rid \)i, by INSTRUMENTS OF NAVIGATION. 93 the menns sliown, there remains the deviation caused by the variable force. This force comes from the transient maj^netisiu of horizontal iron ; or, from the transient magnetism induced by the earth's horizontal force, in iron in any position. It causes a deviation which has four equidistant zero points, and is alter- nately easterly and westerly, in the intervening quadrants. It is for this reason called Quadrantal Deviation. The follow- ing diagram will show how it is caused, and why it takes that fortu. Let the figures in the diagram represent the nppor deck of a ihip, in the several positions, and the fore-and-aft and thwartship lines thereon represent the horizontal magnetic axes of the ship, passing through the centre of the compass. Considering the magnetism of these axes to be positive in the ends presented to the north, it will be seen that, with the ship's head north, there will be no deviation; with the head N.E., the thwartship magnet- isti) tends to deHect the needle to the riglit, while the fore-and-aft 94 NAVIGATION. magne'tism tends to deflect it to the left. From the proximity of the poles of the thwartship magnetism, as compared with the poles of the fore-and-aft magnetism, the deviation is always easterly. It is as well, however, for the student to recognise the possi- bility of its being westerly, as in the ease of a very flat vessel, where the compass might be placed not much above the screw- shaft, or keel. Following the vessel round in the several positions, it will be fseen that there is a deviation, alternately easterly and westerly, having its zero points when the shiji's head is on the cardinal points ; and that there is always a diminution of the pointing force of the needle. No. 242 shows, that a qnadrantal deviation of this kind could be corrected, and the pointing force of the needle increased, by placing iron on each side of the compass, directly athwartships. The compass might be so placed, with reference to the iron about it, especially if it were out of the middle line of the ship, that the magnetic axes would be oblique in the ship. In that case, the zero points of the quadrantal deviation would not be at the cardinal points. No. 242 shows that any quadrantal deviation can be corrected, by placing iron beside the compass, at the same angle from the ship's head, as the zero points which have easterly quadrantal deviation on their left, are from the north point of the compass. 250. Besides the semicircular and quadrantal deviations, there is sometimes a residual deviation, which has the same value in whatever direction the ship's head may be, and is therefore called the Constant Deviation. No. 243 shows that if a compass were jjlaced near iron, such as bulkheads, in somewhat the relative position of the corrector there shown, a positive or negative constant deviation might be caused, and that either one or the other can be corrected, by correctors placed tangentially. 251. Reverting to the force which is in some fixed direction in the ship, the deviation caused thereby must be the same in amount, but contrary in sign, when the ship's head is in opposite directions ; or, when the deviation is small, in opposite directions by compass. Looking at the cause of the variable force, whatever may be the position of iron about a compass, that force will be the same when the ship's head is in opposite directions. The deviation caused thereby will also be the same in amount, and have the same sign, when the semicircular deviation has been corrected, or is small. These facts show how the deviation caused by the variable force may be separated from that caused by the constant force. Let the deviation on each point of the compass be tabulated iu the following form : INSTRUMENTS OF NAVIGATION. 9.1 Head Deviatioa Head Deviation Column I. Column n. Column III. North O South ii E. U E. oosW. 1 57 E. N. bE. 420E. S.hW. 8 30 E. 6 25 E. 2 W. 2 12 E. N.N.E. 640E. SS.W. 8 30 E. 735E. 3 35 ^^ 2 E. N.E. h N. 8 50 E. s.w.bs. 7 40 E. 8 15 E. 4 25\V. 1 55 E. N.E. 10 E. sw. 6 E. 8 E. 4 05 W. I 57 E. N.E. b E. 10 20 E. S.W. b w. 3 10 K. 645E. 2 50 W. I 57 E. E.N.E. 9 50 E w.s.w. 20 w. 4 45 K- 40 W. 202E. E.bN. 8 :oE. W.bS. 3 50 W. 2 10 E. I 40 E. 155E. East 7 E. Wfst 7 10 W. 05 W. 8(15 55 E. 1 E. b S. ESE. S40E. 4 30 K. W bW 940W. 11 40 vv. 2 W W N.W. 3 35 W. Constant |2 E. S.E.E. SE. 3 40 E. 3 40 E. N.W. b W. N.W. 12 30 W. II 50 w. 4 25 W. 4 05 W. deviation S.E. b S. 40 E. N.W.bN. 9 40 W. 2 50 W. S.S.E. 5 30 E. N.N.W. 6 50 W. 40 w. S.bE. 6 soE. N. b W. 330W. I 40 E. Tate lialf the sum of tlie deviations, on each pair of opposite points, and insert it, with its proper sign, in column T. From what has gone before, this must be the deviation caused by tlie variable force, on each of the two points. That is, on the nortli and south points, there is 4° easterly deviation, on N. b E. and S.bW., 6° 25' E., on N.N.E. and S.S.W., 7° 35' E., from that force. So the deviation caused thereby can be ascertained on every point of the compass. To find how much of column I. has the same valne on every point, bring up its lower half into column II. Insert half the sum of the values in columns I. and IT., with its proper sign, in column III. Each value in this column will be that of the meau of the deviation on four points 90° apart, and should be equal to each other, and to the mean constant deviation 2° 0' E. The deviation in column I., made up of the quadrantal and the constant deviations, has the same value in all parts of the world. Because, the disturbing force and the pointing force of the needle vary together, both depending on the earth's hori- zontal force. It also changes but little with time, losing about •05 of its value in a year, owing to the fact that iron slowly loses its capacity for receiving magnetism by induction. It may be worth noting here, that this quantity has nearly the same value, at compasses similarly placed, in ships nearly alike. The correction by soft iron is also perfect for all time and places, if the magnetism of the correctors is derived from the earth's force only ; but when the correctors are placed so near to compass needles, as to receive magnetism by induction from them, though it adds to their power as correctors, the correction ia to that extent imperfect, the correctors having less effect when the horizontal force is increased. The soft iron correctors should ou no account be less than the length of the needles, from their ends. Vti NAVIGATrON. 252. WLen a compass is placed on the upper deck, in tlie middle line of the ship, with tlie iron in about the same relative position on each side of it, and the usual height for taking bear- ings, the maximum quadrantal deviation is about 6° in a new iron ship. Its zero points are at the cardinal points, and there is no constant deviation. In compasses placed in very unfavourable positions, the constant deviation has amounted to 12°, and the quadrantal deviation to 24°, and possibly more. It is not customary to con-ect the constant deviation by soft iron, as it occurs generally only in compasses not required for taking bearings. To meet it, the binnacle, or the compass in the binnacle, or the lubbei--line itself, is so placed, that it points the value of the constant deviation, on the starboard bow, when positive, and on the port bow when negative. Thus, a course steered by a compass, having the lubber-line so placed, is unaffected by the constant deviation. If the quadrantal and constant deviations were not corrected, or were only partially corrected, column I. (251), the sum of their values might be tabulated on each point of the compass, whenever opportunities occur of swinging the ship completely round. Bearing in mind what has been said (No. 251), this quantity should soon become very exactly known, leaving only the semicircular deviation to be ascertained. 253. The horizontal forces causing the semicircular deviation, are best considered as resolved in the fore-and-aft and in the athwartship directions. The fore-and-aft force causes a maximum deviation, when the ship's head is east or west. The athwart- ship force causes a maximum deviation, when the head is north or south. Looking at the deviation table (No, 251), and allowing for the value in column I., it is evident that in this case there is a force towards the ship's head, capable of producing a maximum deviation of 7° 5', and that there is a force towards the ship's port side, capable of producing a deviation of 4°. Therefore, to adjust this compass, a force must be produced by magnets, or Flinders' bar, or both, towards the stern, leaving 5' westerly deviation on the east and west points ; and towards the starboard side, leaving 4° easterly deviation on the north and south points. These residual quantities must be corrected by the means already explained, Nos. 249, 250. Hence the law for correcting the semicircular deviation. Make the deviation zero on any two adjacent cardinal points. If it is known, or, from the position of the iron about the com- pass, suspected, that there is deviation on those points from the variable force, then the ship's head must be placed on the oppo- site cardinal points also, and half the deviation found thereon taken out. 254. The question naturally arises, as to how much of the semicircular deviation should be taken out by magnets, and how IXSTntTMKXTS OF NAVIOATION'. 97 much by Fliiiclors' bar. At first, tliere is no other f^viide tliau the position of the compass (No. 247) ; but when a ship has gone into positions where tbere is much chanp^e in dip and horizontal force, a better jud<,fmentcan be formed. At the magnetic equator, there can be no transient magnetism in vertical iron ; all the semicircular deviation there found, must be caused by tlie ship's permanent force. Hence if, near the magnetic equator, the semi- fiircular deviation be corrected by magnets, any deviation sub- sequently found, arising fi-om change of place, should be corrected by Flinders' bar. From the fact (No. 246) that one part of the semicircular deviation varies inversely as the horizontal force, and the other part as the tangent of the dip, the value of each of these parta can by ascertained, if the deviation is observed in two magnetic latitudes. Example. — The steamship Scotia, having a standard compass in position d (No. 247), corrected by magnets in the Thames, soon after, in latitude 30° S., longitude 10° E., found 12° easterly deviation on the east point, and 10° westei-ly on the west point. How much of the deviation on those points should be corrected by Flinders' bar? From map 232, and table 233 : — Thames, dip 67}°; Nat. tan. of dip 242 ; Hor. force -48. l2u.^6°E.}'^'P - 5'°' N"'- tan. of dip - i 24 ; Hur. force ■K4. (I) Thames . £_ + T x 2 42 = o. ■45 ^'^ Un ^6° E } ^ ■•■ '^ " ~ ' '"'* " "° "•">''^»'«"1»' »° e»»t 1™"''- From(i)P= -116T, Wibstituting in (2)- 3 39 T = 11° semicircuUir on east point. Therefore T= -'' = -3-25° on east iioint. lu the Thames . — 3-25°x 2-42 (the t.in. of dip)= -7J°. Lat 30° S., Lon. 16° E. — 325° x - l 24 (the tan. of dip) = 4°. Therefore, a Flinders bar should be placed before the compass, capable of deflecting the needle 7|° in the Thames, and 4° at the southern position. These deflections, from the magnetism of the bar, will be in opposite directions, and will exactly correct the deviation caused by the transient magnetism of vertical iron. Clearly, the magnetism of the funnel, in this case, draws the north point of the needle aft, in north magnetic latitude ; and forward, in south magnetic latitude. A convenient form of Flinders' bar is fitted to the binnacle of Sir Wm. Thomson's compass. 25.'). The value of the semicircular deviation, on the east or west point, is a key to the value of the deviation caused by the H J)8 NAVIGATION. force in the fore-and-aft line, on every point of the compiiss. Similarly, the value of the semicircular deviation, on tlie north or south point, is a key to the value of the deviation caused by the force in the thwartship line, on every point of the compass. Aa the deviation on any point is made up of that caused by the forces in these two directions, added to that caused by the variable force, it is evident, that if the latter be known (No. 252), and the semicircular deviation be ascertained on two adjacent cardinal points, the deviation table can be completed. When the semicircular deviation is small, the following tab'« will be useful for that purpose : — Semicircular deviation cruised by the san 8 force, on each point, Semiclrcnlar recliouea n gilt ajid left from that car Ural poi nt, through drTiatiun on tlie aJjacen t quadra " point 1ft 2nd 3rd 4th 6th 6tb 7th 8tb 1 „ o . „ , / , , . ( I o 59 55 SO 42 33 23 12 2 I 58 I 40 I 25 I 7 46 23 3 2 57 2 46 2 30 2 7 I 40 I 9 35 3 55 3 42 3 20 2 50 2 13 I 32 47 5 4 54 4 37 4 9 3 32 2 47 ' 55 59 6 5 53 5 33 4 59 4 15 3 20 2 18 I 10 7 652 6 28 5 49 4 57 3 53 2 41 I 22 8 7 5' 7 24 6 39 5 39 4 27 3 4 I 34 9 8 50 8 .9 7 29 6 22 5 00 3 27 I 45 »o 948 9 '4 8 .9 7 4 5 33 3 50 I 57 Example. — The deviation (table 251) having been observed to be 8° 0' E. on the south point, and 7° 10' W. on the west point, what is the deviation on the N.W. b W. point? The semicircular deviation on the south point, allowing for the value in column I., must be 4° E. It is therefore 4° W. on the north point, and, from the above table, 2° 13' W. on N.W. b W., five points from north. The semicircular deviation on the west point must be 7° 5' W., it is therefore 5° 53' W. on N.W. b W., three points fi-om west. Therefore the whole deviation on N.W. b W. must be 2° 13'W. + 5° 53'W. + 4° 25' W. (the value in col. !.) = 12° 31' W. The semicircular deviation being the same in amount, with contrary signs, on opposite points, the deviation on S.E. b E. is 8° 6' E. + 4° 25' W. = 3° 41' E. In the same manner, the devia- tion on every point of the compass can be estimn.ted. There may be circumstances where it would be convenient to ascertain the position of the correctors necessary to apply to a compass, by measuring hor. force (234). The most simple way of looking at the problem is, to consider a ship lying with her head in any known magnetic direction. By placing a horizontal magnet at right angles to the compass needle, and so keeping it, the needle may be made to stand in the direction of the magnetic ixsti;umi;nts of navigation. 99 mcridiai). By placing another horizontal niagni't in the line of the magnetic meridian, the force with which the needle points iriaj- be made equal to the force on shore. Thus, all the forces due to the ship's magnetism, may be counteracted with the ship's head in the one direction. But when the ship's head is moved round, the needle will move away from the magnetic meridian, by reason of the change in the variable force. When the head is in the opposite direction, the deviation will be nearly equal to twice that caused by the variable force, and the needle will point with a force which will differ from the horizontal force on shore, by twice the value of the component of the variable force in the direction of the needle, nearly. Therefore, to counteract the force which causes the semi- circular deviation, the distance of the magnets from the card must be so adjusted, that the needle points with the mean value of the force found with the ship's head in the two directions, and with half the deviation found in the second position. Another way of dealing with the problem is suggested by considering the following facts. If the force with which the needle points is the same when the ship's head is east and west, there can be no constant force in the athwartship line. If it is the same when the ship's head is north and south, there can be no constant force in the fore-and-aft line. Therefore, when these conditions are fulfilled, there can be no semicircular deviation. Further, if the foi-ce is the same on the four cardinal points, there can be no quadrantal deviation. Working by force is a more delicate operation than working by bearings, and, under the circumstances in which the seaman lias generally to work, is scarcely capable of the same degree of accuracy. If advantage be taken of the known direction of docks, wharves, transit and other lines, there will be few occasions where it will be necessary to have recourse to measuring force. But with the two methods available, there should be no detention of ships in port for the purpose of compass adjustment. 256. Hitherto the effects of the vertical component of the ship's forces have not been considered, because a vertical force cannot deflect the compass-needle, right or left. But when a ship heels, a force previously vertical may be no longer so, and the position of the iron about a compass n)ay be so changed, as to introduce a new magnetic force. The deviation, caused by this change in a ship's magnetic forces, is called the Heeling Error. To estimate or correct the heeling error with theoretical accuracy is not an easy problem ; especially in certain positions in a ship, and with the semiuirc'ular deviation uncon-ected. The following remarks must be considered as applying to a compass, in such a position iia is usually selected for a standard compass, and having the semicircular deviation corrected. At a compass so situated, there will be a force upwanlB or dcjwnvviU-ds in the ship, caused by pei'- 1 00 NAVIGATION. maiient magnetism. The value of this force will depeiid, maiiilr, upon the direction in which the ship was built, and the position of the compass in the fore-and-aft line. It may be counteracted by a magnet placed end on (239), and vertically below the centre of the compass. If it is not counteracted, it will, by coming partly on one side when the ship heels, draw the north point of the compass to one side or the other. There will also be a force upwai-ds or downwards in the ship, fioiu the transient magnetism of vertical iron, depending for its value on the earth's vertical force, of which it is a constant fraction. This force, in north magnetic latitude, is that of a nega- tive pole under the compass, changing to positive in south magnetic latitude, drawing the north point of the needle to the high side of the ship in the former case, and to the low side in the latter. This force evidently should not be counteracted by a fixed magnet, but by a bar of soft iron, having, in noi'th magnetic latitude, negative magnetism in the end nearest to, and above, the compass. "05 of tlie earth's vertical force is about a mean value of the vertical force caused by induction therefrom ; therefore, in correcting the heeling error by a vertical magnet, the vertical force of the earth and ship should be brought to about 1'05 of the earth's vertical force, wherever the ship may be. Sometimes the position of the funnel, or an iron mast, may be such, that its vertical transient magnetism counteracts that of the ship ; this will probably be the case in a compass in such a position as d (247). Or it may be counteracted by putting the upper end of the Flinders bar, where one is used, above the level of the compass. Looking at the magnetic condition of athwartship iron, such as beams, passing under the compass, when, from the ship heeling, it departs from the horizontal position, it is evident that the higher ends will liave negative magnetism, drawing the north point of the compass-needle to the high side of the ship in north magnetic latitude. The reverse of this takes place in south magnetic latitude, therefore this force should not be counteracted by a fixed magnet. If a soft iron bar were placed horizontally athwartship, on each side of the compass, the magnetism induced therein would, if they were of suitable size and distance from the compass needle, exactly counteract the magnetism induced in the athwart- ship iron of the ship. This condition is nearly fulfilled by soft iron so placed as to correct the quadrantal deviation, so that no separate corrector is required for this part of the heeling error. Because the transient magnetism of horizontal fore-and-aft iron, below the compass, causes a vertical force which is zei'O when the ship's head is east or west, it is desirable to correct the heeling error when, the ship's head is nearly on those points. Then, if the quadrantal deviation is corrected, and the vertical INSTRUMENTS OF NAVIGATION. 101 force brought by a maijnet to the same value as, or a little niora than, the vertical force ou shore, the heeling error will be practi- cally corrected. The forces which cause the heeling error, by drawing the north end of the needle to one side or the other, must have their maximum effect when the ship's head is north or south. When the ship is rolling, the north end of the needle being drawn to each side alternately, causes the card to be unsteady. This dis- turbance of the eouipass-card has probably been more trouble to the navigator, than the error produced by heel. Thus, in dealing with compass deviation, there are two distinct problems : one, to ascertain its amount ; the other, to get rid of it altogether. At first sight, one or the other of these processes appears unnecessary, and in the early days of iron ships some thought that, with a table of deviation, there was no need for correctors ; others that, if the compass were corrected, there was no need for a table of deviation. Experience has long since shown that neither of these views was correct. Many iron ships could not be navigated unless the compass was, at least, partly corrected. On the other hand, though compasses are frequently BO well adjusted as to be without deviation, there are small changes subsequently which cannot be safely disregarded, render- ing a deviation table necessary. Changes which are gradual can be met by the ordinary daily observations, which should never be omitted ; but there are some changes which are sudden, against which the seaman must be on his guard. If a ship has been steering for some time on one course, she will acquire negative magnetism in the part of the ship towards the south. On first altering course, the north point of the compass is likely to be drawn, for a short time, towards that part of the ship which was previously south. This is espe- cially the case in changing from courses near east or west to those near north or south. Of course, the same effects follow when a ship has been some time in dock. Thin iron structures, such as funnels, funnel casing, or venti- lating cowls, are liable to change their magnetic state from strains or concussion, and so affect the deviation ot a compass placed near. Any shod: or strain which causes iron to vibrate or bend, and so cause movement in its particles, facilitates magnetic change. With the introduction of electric lighting on board ship, came a new form of compass disturbance. The magnetism of the large electro-magnets, in the dynamos at present used, may disturb a compass at the distance of sixty feet. Also, circling round the wires conducting electricity, and at right angles to their direction, is a magnetic force, going in one direction round the wire conducting the direct current, and in the opposite direction round the wire conducting the return current. Thus these forces counteract each other when the conducting wires 102 navigatio:t. uie together, but when they are separated cause a propo-i'tionnl Jisturbauce to the compass. The maximum value of this disturbance, for any speed of the dynamo, is apparent directly the dynamo ia started at that speed. So, by starting and stopping the dynamo, with the ship's head on two adjacent cardinal points, and noting the eifects, the value of the disturbance on every point of the compass can be ascertained. Table 255 will be useful for this purpose. 257. A method of measuring the effects of a ship's magnetic forces, in causing deviation, was introduced by the late Mr. Archibald Smith. He found that the deviation could be expressed, as in the following equation : — Deviation with ship's head on , j^ j ^. ^ . cos f + D . sin 2 (' + E . cos 2 C any point . . . . ) ' ' * ' The factors A, B, C, D, E, are called coefficients, and f is the direction of the ship's head by compass, reckoned round the cii'cle to the right. Therefore, in dealing with the equation, the sea- man, who generally has to deal only with angles not greater than a right angle, must consider the sign of the direction of the head, as well as that of the coefficient, in each term. A, the first term in the expression, is the value of the con- stant deviation (250). It may be found by taking the sum of the deviation on the four cardinal points, and dividing it by four. B is the maximum value of the deviation caused by the force in the fore-and-aft line (253). It is + when the force is towards the ship's head, and — when towai-ds the stern. It maj' be found by adding to the deviation on the east point, the deviation on the v.-est point with its sign changed, and taking half that sura. Any constant force in the fore-and-aft line, which causes this devia- tion, must cause a deviation = B . sin f, the second term of the expression, on every point of the compass. C is the maximum value of the deviation caused by the force in the athwariship line (253). It is 4- when the force is towards the ship's starboard side, and — when towards the port side. It may be found by adding to the deviation on the noi-th point, the deviation on the south point with its sign changed, and taking half that sum. Any constant force in the athwartship line, which causes this deviation, must cause a deviatiou = C . cos 5", the third term in the expression, on every point of the compass. D is the mean value of the deviation on the inter-cardinal points, caused by the vaiiable force (249). It may be found by adding to the deviation on the N.E. and S.W. points, the devia^ tion on the S.E. and N.W. points with the sign changed, and taking the fourth part of that sum. A force varying regularly, and causing this deviation, must cause a deviation = D . sin 2 f, the fourth term of the expression, on every point of the compass. £ is the mean value of the deviation on the cardinal points, 1^TSTKUMI■:NTS OF NAVIGATION. 103 can?etl bj the variabK' force (21-9). It may be found by adding' to the deviation on the north and south points, the deviation on the east and wast points with the sign changed, and taking the fourth part of that sum. A force varying regularly, and causing this deviation, must cause a deviation = E . cos 2f', the fifth term of the expression, on every point of the compass. The existence of the E shows that the axes are oblique (249). It is obvious that the foregoing statement of the effect of the forces in causing deviation is true only when each force is the only disturbing force on the needle ; it is true enough when those forces are small : in that case the resulting deviation is also small, and the sum of the five terms is equal thereto ; when the deviation is large, the coefficients must be determined with more exactness. With such deviations as are usually found, since the general adoption of compass adjustment, the method here given is sufficiently exact. The student must not consider the coefiicients as forces, or as in any way causing the deviation ; they merely measure it, with more or less exactness. And by their means the parts of the deviation can be particularised, in speaking and in writing, and a record of its value kept in five terms, of which two are generally zero. Excepting for this purpose, the treatment of the subject by coefficients, especially laborious methods of determining their exact values, and of deriving the ship's magnetic forces there- from, has never been greatly esteemed by navigators. 258. Professor Airy made use of the terms Eed and Blue, to indicate the two kinds or states of magnetism, of the nortli and south ends of the compass needle respectively. These terms have been of great use, especially in making clear, by coloured diagrams, the distribution of magnetism in iron ships. The terms positive and negative have been used in this chapter, being in accord with the terms used in the kindred science of electricity, which is daily becoming of more importance to seamen.* The subject of compass deviation and adjustment was tho- roughly investigated by a body of scientific men, shipowners, and others, interested in the subject, called the Liverpool Compass Committee. The results of their labours were published, in language intelligible to seamen, in three most valuable reports to the Board of Trade, 1856, 1857, 18GI. • I'rofessor (now Sir) George Biddell Airy, K.C.B., lias lived t.) see bis accurate and tlioroughly practical method of iidjiisting compasses, diviscd half a century ago, over- come all o|ipo>itiun, and he now, and for many years pas-t, uuiversally adopted. He has in other ways furthered ihc science of navigation, but in I'aciliiaiiDg the navigation o( iron ships he is pre-eminent. KxVVKiATION. ITie following Notes are the result of recent theory and experience. The numbers refer to Articles in the jfyresent edition. Alt. 215. The method of suspension by india-rubber has been discontinued, owing to its rajiid deterioration when exposed to heat and wet. 216. In Lord Kelvin's (Thomson) compasses the outer graduation of the numerals is inverted in the Navigational or Standard Compass to enable the card to be read direct with the azimuth mirror. The average period of a Thomson's card varies from thirty seconds for a ten-inch card to thirteen and a half seconds for a four-inch one. 219. The Variation at Greenwich was (1899) 16° 34' westerly, de- creasing 7' annually. 220. The simultaneous appearance of auroras and disturbances of the magnetic needle (magnetic storms) are manifestations of the same cause. The late Father Secchi held that thunderstorms exer- cised a perceptible influence on the magnetic needle. The disturbing element of land on the compass needle is recognised to be submarine. Theory confirmed by experience show that if the rocks are the upper extremities of a ridge in north magnetic latitudes they would attract and in southern repel, the red (paragraph 239) end of a compass needle. 223. The prefix correct to true, magnetic and compass courses is being discontinued, a true course is a compass one corrected for varia- tion and deviation ; a magnetic course, the same corrected for devia- tion, and a compass course, one uncorrected for variation and deviation. 232. The Dip of the needle at Greenwich was (1899) 67° 10', de- creasing \'l annually. 237. The expression, " magnetism by induction from the earth " is seldom used ; the magnetism of both earth and soft iron are produced by the same lines of magnetic force. 239. To avoid ambiguity, the pole of a magnet that attracts the north-seeking end of the needle is called blue and the repelling one red, bearing in mind the pole in the north end of a compass needle is a true south pole, and that in the south end of a compass needle is a true north pole. 244. Read paragraph at 237. Gaussin error is often developed by magnetic induction in a ship's iron beams, more especially when ])rOf>eeding east and west ; in fast Atlantic liners a Gaussin error of 8° to 10° is not unusual daring a voyage across the Atlantic. 249. A compass is usually corrected in the following order : the quaflrantal error, the heeling error, and lastly the semi-circular error. 256. In merchant vessels arrangements are usually made to place mSTRtlMENTS OF NAVIGATION. IOIJB the navigational compass beyond the magnetic field of the dynamo, but the necessaiy arrangements in a man-of-war may prevent this being carried out. A compass if within the magnetic field of a dynamo will be disturbed, the error altering with change of azimuth.' In the general tj-pe of dynamo suppUed to H.M. ships, designed for 80 volts at the terminals, the minimum distance of a compass should be 60 feet from a 300-ampere machine, increased to 70 feet from a 400-ampere one. A 600-amp6re machine being armour-clad and multipolar produces no disturbance on a compass 15 feet away. In the " Destroyers " the con-ection is made by an electro-magnet at the foot of the compass pedestal, with its poles reversed to those of the dynamo ; in second class cruisers (Apollo class) by exciting the shunt coils of both dpiamos, when only one is in use, the resulting disturbances are neutralised, provided the poles of the dynamos are gymmetrical to the middle line. In the electric lighting of a compai^s, the current is usually con- veyed to a 16 c.p. lamp by a twin cable, protected by phosphor- bronze braiding. The best po.sition is to place the lamp vertically above the axis of the compass needle ; occasionally a disturbance arises from the inductive effect due to the current in the filament of the lamp itself. A small electric light (half-candle power) is found useful for star azimuths at night or if fitted to a sextant for stellar observations. * For detailed information see The Hfrtriner'i Covipaas in an Iron S/iip, by (Captain J. ^Vhitly Dixon, B.N., sold by J. V. Potter, 146 Miooriei, Lun.iuu, K 104 KA VI G AXIOM. 11. The Log and Glasses. 1. The Log. "J59. The log consists of the log-ship and line. The log-flliip is a thin wooden quadi-ant, of about five inches radius ; the cir- ciihir edge is loaded with lead, to make it float upright, and at each end is a hole. The inner end of the log-line is fastened to a reel, the other is rove through the log-ship and knotted ; and a piece of about eight inches of the same line is spliced into it at this distance from the log-ship, having at the other end a peg of wood, or bone, which, when the log is hove, is pressed firmly into the unoccupied hole. At ten or twelve fathoms from the log-ship a bit of buntin rag is placed, to mark off a sufficient quantity of line, called stray- line, to let the log go clear of the ship before the time is counted. 260. The log-line is divided into equal portions, called knots, at each of which a bit of string, with the number of knots upon it, is put through the strands. The length of a knot depends on the number of seconds which the glasses measure, and is thus determined : The No. of feet in I knot : No. of feet in i mile : : No. of seconds of tlie glass : 3600 (ihc No. of seconds in an liour). The nautical mile being about 6080 feet,* we have, for the glass of 30 seconds, the knot = -3^° = 50-7 feet, or 50 feet 8 inches, for the glass of 28 seconds, the knot = ^^^ = 47-3 inches, or 47 feet 4 inches ; and so for any other glass. 261. The knot is supposed to be divided into eight equal parts, or fathoms (which they are very nearly). In the Royal Navy the kjiot is divided into tenths and the even fathoms only are reckoned, for the convenience of adding up the distance on the log-boa rd.f 262. The log-line should be repeatedly examined, by comparing each knot with the distance between the nails, which are (or should be) i^hieed oti the deck tor this piirpose, at the proper distance. The line should be wet whenever it is required thus to remeasure it, or to verify the marks. * The Gengiaphical Mile is generally defined to be tlic lentrih of a niinnte of ir? 1:2 iVc eartli's eqnjtor ; but the Nautical Mile as defined by hydiographers is the length of a minute of the meridian, and is slightly difterent for every ilitferent latitude. (6« Table 64a.) It is equal to a minute of arc in a eirile, whose radius is the radius of the curvature of the met idian, at the latitude of the place. t It is, of course, more systematic to divide the knot or mile into tenttis, ns in th« Traverse Ta'ile, instend of eighths ; btit Muyle tenths and fathoms may be used for each olhfs without sensible error. INSTRUMENTS OF NAVIGATION. 105 '26'^. As t[\e manner of heaving the log must be learned at sea. it is onl^ necessary to remark, for reference, tliat the line is to be faked in the hand, not coiled ; that the log-ship is to he thrown out well to leeward to clear the eddies near the wake, and in such a manner that iv may enter the water perpendicularly, and not fall flat upon it ; and that before a heavy sea the line should be paid out rapidly when the stern is rising, but when the stern is falling, as this motion slacks the line, the reel should be retarded. 264. (2) Massei/s fjog shews the distance actually gone by the ship through the water, liy means of the revolutions of a Hy towed astern, wliith are registered on a dial-plate. This log is highly approved in practice.* •235. When the water is shoal, and the set of the tides or current much atTected by the irregularity of the channel, or other causes ; and when, at the same time, either the ship is altogether out of sight of land, or the shore presents no distinct objects by which to fix her position, recourse may be had to the ground log. This is a small lead, with a line divided like the log-line ; the lead remaining fixed at the bottom, the line exhibits the etfect of the combined motion of the ship through the water, and that of the water itself, or the cur- rent ; and therefore the course (by compass) and di.stance made good are obtained at once.f Caution. — Logs, whether patent or common, are unsatisfactory instruments in these days of high speed. No patent log yet in- vented will stand the wear and tear of a fast ship for any length of time. To avoid this wear and tear they should be used only when coasting or in with the land. They will tell a different story in a head sea to what they do in a following sea. In slow steamers and sailing ships they are naturally more reliable. Still, logs must be used ; but it must be remembered they are beset with ivipedimerds, and their indications must not be implicitly trusted in critical times. Ry practice, seamen learn to estimate the rate of progress of the sliij) closely by the number of revolutions in a given time made by tlie engines; but this is only speed through the water ; the sailor has to consider carefully what that unstable element has also been doing. J Further, though ships may now better preserve a given course, and the di.stance run may be estimated more accurately than formerly, there are in modem iron ships elements of uncertainty about D.R. which still makes it perilous to close the land unless there are means of knowing with some certainty the ship's proximity thereto, especially where land has a bad reputation, as Ushant, C. Kinisterre, C. (iuardafui. Mocha I. in South America, &c. * Other logs on tins principle hftve since been invented and are in common use: DoMl.ly, Walker's laffrail log. They should be well oiled, and stowed away dean. t in numerous passages up and down the rivor Plate, whore the above circumstances ccnriir. Captain 0,,rd.,ii T. Falcon, in 1818-I9-20, made constant use of this log. J Stt Admiralty Current CliarLs, Tide Tables, and Sailing I/ip ctioiie, Nos. 951, 1)52. lUG NAVIGATION^. 2G6. (3^ The Glauses. — The long glass runs out in 30' or in 28"; the short glass runs out in half the time of the long one. When the ship goes more than five knots, the short glass is used, and the number of knots shewn is doubled. 267. The sand-glasses should frequently be examined by a seconds watch, as in damp weather they are often retarded, and sometimes hang altogether. One end is stopped with a cork, which is taken out to dry the sand, or to change its quantity. 268. When either the line or the glass is faulty, or when a line and glass not duly proportioned to each other are employed, the distance run is found as follows: — The number of feet in l*" is to the number of feet run out in an observed number of seconds, as 3600 (seconds in an hour) are to the observed number of seconds. Ex. Suppose 190 feet of line are run out in 22* : required the rate. The number of feet run out in l"" : 190 :: 3600" : 22'; hence the number of feet = L22Ji3529 = 31090 feet; which, divided by 6000 (as near enough), gives S'Z miles. CHAPTER III. The SailIxNGS. I. Plane Sailing, with Tkaverse, Current, and Windward Sailings. II. Parallel Sailing, with Middle Latitude, AND MERCATOR'S SAILINGS. III. GrEAT CiRCLE SAILING. 269. In considering the place of a ship at sea, with reference to any other place which she has left, or to which she is bound, these five things are involved : the Course, Distance, Difference of Lati- tude, Departure, and Difference of Longitude. 270. In practice these two general questions occur. 1st. The course and distance from one place in given latitude and longitude to another are given, and it is required to find the latitude and longitude of the otlier place. 2d. The latitudes and longitudes of two places are given, and it is required to find the course and distance from one to the other. The methods of solution, that is, the rules of calculation, by which the answers to such questions are obtained, are commonly termed Sailings. I. Plane Sailing. 271. In Plane Sailing, as the term implies, the path of the ship is supposed to be described on a plane surface. If the ship sails 1 mile on a given comse, she makes a certain D. lilt, and Dep. ; in sailing a second mile, on the same course, she THE S.\tLIN(.8. 107 imkf'rt good tlic' isaiiif 1). lar. aiifl Dep. as before. Thus the D. lat. Kiiil I)e|j. for 2 miles of Di>t. are twice those for 1 mile; for ;J miles of Di-^t. tliey are three times those for 1 mile, and so on ; that is, the total D. lat. and Dep. made good are proportional to the Dist. on the s|)hcre as they would be on a plane. Plane Sailing, accordingly, treats of the relations of the Course. Dist., D. lat., and Dep., and applies to right-angled triangles generally. But each mile of Dep. which the ship makes good corresponds to a Diff. of Long, which is different according to the latitude in vhich the ship moves (Note, p. 68), that is, there is no comstunf. pro- portion between the Dep. and Diff. Long, in two different lati- tudes, and therefore a question in which Diff. Long, is concerned is not within the province of Plane Sailing, except the case in which the ship is on or near the equator, where Dep. and D. Long, are the same thing. 272. (1.) The proportions, No. 162, p. 46, as adapted to the figures, No. 200, p. 59 (or to the third figure of No. 163, where the course is the angle ABC), give the proportions or canons, as they are called, of Plane .Sailing, ^^'e employ the following: Disf. : Dej.. : :: rad. ( = .) : 8in. Co., whence, Dep. = Dist. x sin . Co. (»•) Uist. : D. Lat. : I : COS. Co., D. Lat. = Dist. X cos .Co. (20 . I-at. : Dep. : 1 : tan. Co., Dep. = D. Lat. x tan . Co. (3.) . U: . DUt. : 1 : sec. Co., Dist. =D. Ut. xsec Co. (5.) (6) (a.) These equations put into logaritlims by the rules Nos. 64 and 55, p. 20, become Log. Dep. = log. Dist. + log. sin. Co. — lo (1.) Log. D. Lat. = log. Dist. +log. cos. Co.-io (2.) Log. Dep. = log. D. Lat. +log. tin. Co. — lo (3.) I-og. tan. Co. = log. Dep. + lo — log. D. Lat. (4.) Log. Dist. = log. D. Lat +log. sec. Co.— lo (5.) Log. sec. Co. = log. Dist. + lo — log. D. Lat. (C.) Which logarithmic equations contain the rules employed. On ordinary occasions four places arc enough. Case I. Given the course and distance, to tiiid the difference of iBtitude and departure. F.i. 1. .\ ship sails N.W. by N. 03 miles from lat. 49'' 30' N. ; find the D. Lat. and Dep. and also tlie Lat. in. 278. By Tn.fpecfion. Open Table 2 at 3 Points,* and against Hie Dist. 1*03 stand D. Lat. 85-6 and Dep. b7-2. Then 85-6 or 1° 26'-0 added to 49" 30' gives Lat. in 50' 55'-6 N. Whenever Ihc rniir»c i« given in points or divisions of a poinl, it nuist he turned d.gries (213; heluic entering 'I'rnvei-.e 'J'.!!.).; 1. If)X NAVrOATTON 27i. Bi/ Compulation. (1.) For (he D. Lat. To tlie log. cos. ot Dhe Oouvse (Table G8) add the log. of the Dist. (Tabk- 6 t) ; the Biiui {rejecting 10 from the index) is the log. of the D. Lat. (2.) For the Dep. To the log. sine of\hc Course add the log. of the Dist..; the sum (rejecting 10) is the leg. of tlie De\t. above. Course 3 points, Dist. 103. cos 9-9198 log. 20li8 log. 1*9326 (Tliis is tlie Canon (i.) in No. 272.) 3 points, or 33 45 Dist. 103 D. Lat. 8 s Course 33^45 Dist. 103 Dep. 57-2 (Tliis is the Canon ( 1 9'744/ 275. By Construction. Draw a line C N towards the nortli for the meridian. From the Lieutre C, witli the chord of 00° as radius, de- scribe a:i arc on the west side of C N, and lay otl" the chord of three ])oiiits, or 33"! to a (No. 107). Through a draw C a, this gives tlie angle N Ca equal to the Course, or three points ; lay off from a scale of equal parts C A equal to the Dist. 103; draw .\ B perpendicular to C N, then C B will shew on the same scale the D. Lat. 85-6, and A B the Dep. 57-2. Ex. 2 .V ship sails Dep., and also the Lat. S. 72° W. 216 miles from lat. 14° 1 1' N. : required the D. LhI. and il Dist. 216 05-4. Bi/ Inspection. The Course 71° and Dist. 216 give D. Lat. 66-7 and Dki Then 66'7, or 1' d'--;, subtruoted from 14° 1 1' N. leaves Lat. in 13° 4'-3 N. By Computation. Course 11° log. cos. 9-4900 I Course 72° log. sin. 9-9782 Dist. 216 loK V3345 Uist. 216 lug- 2-3345 D. Lat. 66-7 log. 1-8245 I t»^-f- ^°5'4 '"g- ^'i'-l 'r^\ By Construction. Draw & line C S to the soutliward for the meridian. By tlie chord of 60° lay off the arc 7 2 ^ to the westward, and draw CA equal to 216; draw AB perpendiciUar to C S, then C B is the D. Lat. 66-7, and A B the Dep. 205-4. These two e.xanijjles of construction are sufficient for all varieties of Case L When the course is to the eastward, C .\ is drawn on tiie riiiht side of the niej'idian C N or C S instead of the left side. Case IL Given the course and difference of latitude, to find tlie distance and the departure. Ex. 1. A ship sailing W.S.W. makes 47 miles D. Lat. : find the Dist. run and the Dep. 276. Bi/ Inxpection. Enter Table 1 with the Course fi points look in the D. Lat. colunm for 47; the nearest to -47 is 47- 1, a<rainst «hich stand the Dist. 123 and Dep. 113-6. TIIK SAII.INT.S. io:> Ih« Ijit. of tlie ship is, from the nature of the case, already ^iveii. 277. By Cnmpiitntion. (1.) For the Dist. To the log. sec. of the Course add tiie log. of tlie 13. Lat. ; the sum (rejecting 10) i§ thb lo^. of tlie Uist. ('2.) For tlie Dcp. To the log. tan. of the Course add tiie log. o( (he D. Lat. ; the sum (rejecting 10) is the log. of the Dep. 6 points, or 67" 30' log. sec. 0-4172 D. Lat. 47 log. 1-6721 Dist. 122-8 log. TxSgj (This is tlie Canon (5.) in No. 272.) Course 67° 30' log. tan. 0-382S D. Lat. 47 log. 1-6721 DeI". 113-5 log. 2-0549 (This in the Canon (3.) in No. 272.) 278. J3j/ Construction. Draw the meridian line CS; lay off the course, or angle 8 C A, 6 points (No. 107); from C lay off C B the D. Lat. 47 ; draw B .-V peruendicular to C S, then C A is the Dist. and A B the Dep. This example will suffice for all varieties of Case IL When the course is to the northward, C N is drawn up- wards insteatl of C S downwards; and when the course is to the eastward, ' C A is to he drawn on the right siile of the meridian instead of the left side. Ex. 2. A sliip sails N. 54° E. and makes 119 miles D. Lat. : required the Distance lun «nd the Departure. By Iiurpection. Course 54° in Table 1, and D. Lat. 119-3, S"'<! 'he Dist. 203 and Dep. 164-2, nearly enough in practice. Case in. Given tlie difference of latitude and departure, to find the course and distance. 279. By Inspection. Look in Table 1 for 91 in the 1). Lat. co- lumn, and 34 7 in the Dep. column; the nearest are 90-6 and 34-8, whicli give the Course 21® (N. 21° E. in this example) and Dist. 97 miles. 280. By Computatiun. (1.) For the Course. From the log. of the Dep. (adding 10 to its index if necessary) subtract the log. of the D. Lat. ; the remainder is the log. lan of tiiC Course. (2.) For the Dist. Find the Course; then to the log. sec. of the Course add the log. of the D. I.at. ; liie sum is the log. of the Dist. "ourse 2o°5i' log. sec. 0-0295 J. Lat. 91 lug. 1-9590 Dist. 97-4 log. 1-9885 n-iiiiis the ( n.oi, 3 ) No. 272.) Dep. 34-7 log. I -540 J D. Lat. 91 log. 1-9590 »E 20" 52' h.| tan. 9-5813 i^ i.s the CaiK » 4.KNO.-272.) no NAVIGATION. 281. Bij Const ruction. Uraiv the ineiidiaii C \. Take C B, the D. Lat. 91, and through B draw Ba perpendicular to C N, and equal to 34-7; join C A ; then BC A, the Course, measures 21° (No. lOfi, 2), and C A, tlie Dist. measures 98. This e.\ample will suffice for all varieties of tiie case. Case [V. Given the distance run and the difference of latitude, to find the course and departure. Ex. A ship sails loi miles between snulh and east, and makes 52 milefl D. Lat : fini the Course and Dep. 282. J3i/ Inspection. In Table 1, 101 in the Dist. column, and 52 in the D. Lat. column, occur over Course 59° (S. 59" E. in this e.\ample), and against the Dep. 860. 283. By Com/mtation . (1.) For the Course. From the log. of the Dist. subtract the log. of the D. Lat.; the remainder is the log. sec. of tlie Course. (2.) For the Dep. Find the Course; tlien to the log. sine of ll.e Course add the log. of the Dist. ; the sum is the log of the Dep. Ex. Dirt. loi log. 20045 l>. Lat 52 log. 1-7160 CoDESE 59^ I log. sec. 0-2883 (This is the Canon (6.) No. 272.) Course 59° i' log. sin. 9'933i Dist. 10 1 log. 2-0043 Dep. S6-6 log. 1-9374 (Thii! i. the Canon (i ) No. 272.) 284. Jiy Construction. Draw the meridian C S. Take C B, tlie D. Lat. 52, and through B draw BA per|)en- dicular to C S. Froui C as centre, with tiie Dist. 101 as radius, describe an arc cutting B A in A ; then the Course, S C A, measures 59°, and B A, the Dep., measures 86 6. This one example of construction will be sufficient. Eji. 1. Ei. 2. El. 3. di. 4. A ship sails from Flanibnrmigh He,u quired her Lat. in, and Dep. Ans. D. La Ejamples far Ejcercue. 54" 7' N., E. by N. i N. 264 •6N., Lat. in, 55''24'N.; D A ship from Lat. 49° 57' N. sails S.W. by W. 244 ni Dep. Ans. La s : required her Lat. in, an IN 47" 41' N. i Dep. 202-' A slii]i sail> S E. by E. from Lat. 1° 45' N., until =he arrives in Lat. o" 31' S. : quired her Dist. and Dep. .\ns. Dist. 244-8 ; Dep. 2c A ship from St. Helena in Lat 1 5° 55' S. sails N.W. \ W. till she is in Lat. 1 3° i find the distance she has run, and the Dep. Ans. Dist. 274-3 I Dep. K». 5. A sbii PS 135 miles northing, and 87-7 miles of Dep. westing: required hei id Dist. made good. An«. Coukse N. 33" \\. ; DiST. 161 n.ilc*. TIIK SAII,IN(iS. ]] 1 B> 6. A ship sMil.i 1 10 miles between N. anJ E., and makes i6o'-9 D. Lat : find the Coui>t mid Oep. Ans. Course N. 40° E. ; Uep. 135 milcg. El. 7. \ ship sails 244 miles between S. and W., and makes 136' D. Lat.; find llie Course and Dep. Ans. Course S. 56° 8' W. : Dei. io>-6. 1 . Resolution of one Course upon another. *28o. It is sometimes required to resolve the distance run upon a given course into the distance upon a proposed course £x. A ship is making good S. 70° W. 5} miles an hour: at what rate ia she neariog • port bearing S.W. } Draw the meiidian, A S, of the ship : of the port, S.W., and the Course S. 70' present the rate per liour (or for a smaller interval), as 54 knots. B then is the place of the ship at the end of this interval. The distancte, A P of the port, being very great, as compared with A B, a circle B D, described from P as a centre, is nearly a right line, and perp. to A P, and cuts off A D, the dist. by which the ship has neared P in an hour. Now A D is the D. Lat. to the Dist. AB, and the angle BAD as Course. BAD equal to "o"' — 45°, or 25°, and Dist. 5^, give AD equal to 5 knots, the rate required, and A D is A B resolved in the direction AP. p " When the number of degrees between the given and proposed courses e.xcecds 90, the ship is increasing lier distance from the port instead of closing it. It is proper to observe, that the ciiange in the distance of the port, made by the ship wiien not .steering- directly for it, is true only i'or its present bearing, and tiierefore iiulds only for a short time. 2. Traverse SaiUiiy. 286. This is a variety of plane sailing in which the ship makes two or more courses in succession. The process of reducing several courses, with the distances run on each, to the single course and distance which the ship would have mac'e good if she had sailed at once from the place she first left, to the place at which she last arrived, is called working a traverse. 287. To work a Traverse. (1.) Draw six vertical lines. Head the space to the left Courses, the first column Distances, the next two columns D. Lat. ; marking the first N. and tiie second S. ; head the last two columns Dep., marking one E. and the other W. This forms a skeleton Traverse Table. (2.) Set down the Courses, and the Distances against them, in order; look out in Table 1, the D. Lat. and Dep. to each Course and Distance. When the ship makes northing (that is, when the Course has an N. in it), set the D. Lat. in the N. column, other- wise in the S. column. When the ship makes easting (that is, when the Course has an E. in it), set the Dep. in the E. column, otherwise in the W. column. \3.) Add the D. Lats. in each column ; write the lesser of the two sums under the greater, atid take their ditt'erence. Do 'lie same with the Departures. 11 '2 MAVIftATlOfK. ('I.; Thesfi differences are the D. Lat. and Dep. made good on the whole, and each takes the name of the column it stands in. The course and distance are then found by No. 279. It may be advisable for a bcoinner, before he proceeds to take out the quantities fioni the Traverse Table, to write a dash in all places not to be occupied by a D. Lat. or a Dep., in order to avoid writing a quantity in the wrong column. The first example only is thus marked, because such helps are useless to an expert computer. Es. A ship sails S.W. by S. 24 miles; N.N.W. 57 miles ; S.E by E. J E. 84 uules i »nd South 3 5 miles : find the Course and Distance made good. Co>,.e, Disl, D. Lat. Dep. N. 1 S. E. 1 W. S.W. by S. N.N.W. S.E.byE.4E. South. ;; 55 5^-7 20-0 39-6 35-0 74-' 13-3 21-8 5^-7 94-6 52-7 74'' 35' 35'' 41-9 39-0 The D. Lat. 41 9 and Dep. 39-0, are found at 43° against the Dist. 57. Hence, since the ship has by the Traverse Table made southing and easting upon the whole, the Course is S. 43" E., and Dist. 57 miles. By Computation. Each portion of the process having already been separately considered in plane sailing, nothing remains to bp added here. 288. By Coiistriiclion. With the chord of 60 describe a circle; draw the meridian N S, and mark the centre C. By means of the scale of chords lay off S 1, ecpial to 3 points, or S.W. by S., for the first course. Lay off N 2, equal to 2 points, or N.N.W., for the second course. Lay off S 3, equal to 5i points, or S.E. by E. i E., for the third course. The fourth course, or south, is already laid off", being on the meridian. Now lay the edge of the ruler on C and on the point 1, and lay off by the compasses, or a scale of equal parts, the first distance, C a, 24. Place the edge of the ruler on «, laying it parallel to the line joining C and the point 2, and lay off the second distance, a b, 57. Place the ruler on the point b, laying it parallel to the line jjining C and the point 3, and lay off the third distance, be, 84. Lay the ruler on c, parallel to the meridian, and lay off cd, the fourth distance, 35. The point d is therefore the place at which the ship has arrived. Join Cd, then SCrf is the course, 43°, and Cd Uiti distance, 57. Also, ilrawiiig Dd perpendicular to OS, give* Tin- SAll.lNCiS. 13 DC the 1). Lat., and I)r/ tlir J^eji. wliieli will be found to measure i; y aiKJ 390. \ \ -y Tliu circle is here drawn outside the traverses altogether, without regard to the dimensions of the scale of chords, merely to shew the process more clearly. This example, after the practice which the learner will have al- ready had in drawing the figures in the preceding articles, will be siitficient for any ease that may occur. Kx. 2. A .hip sails N.N.E. ii miles; N.l miles J N N.W. 4 miles : required the Cours as; E. ^ N. 14 iiiilet ; : made good. WVst, Co„r., r.isi. N. s. E. 1 N.N.E. ,, .0-2 4-i N E. 3 E. 39 23-2 3'-3 E.IN. '4 '■4 ■3-9 West. ■■) '9 N.N.W. 4 3-7 ■•5 38-5 494 20-5 2C-5 28-9 The D. Lat. 38-5 in the N. col., and Dep. 289 in the E. col give Ik.irKSE N. 37" E., Dist. 4H miles. 114 NAVrCATlON. 289. The D. Lat. made good on the whole, as tluis found, boin-j applied to the Lat. left, gives the Lat. in. Thus, suppose in the above example the ship left Lat. 38'' 40' S. ; then :38'-5 northinj; places her in Lat. 38° I' 5 S* Examples for Exercise. Ex. I. A ship from Cape St. Vincent, in lat. 37° 3' N.. 43 mUes, S.E. hy S. 64 miles, and N.N.E. 2 made good, and also lier Latitude in. Ans. Course S. 34° E.-, Dist. 89 miles; Lat. in 35'' 49' N. Ex. 2. A ship from Cape Amber (N.E. extremity of Madagascar), in lat. 1 1"' 5;' S.» sailed as follows : — S S.E 4 E. 33 miles, S.W. by W. 40 miles, S.E. by S 44 miles ; N. 36 miles, S,W. by S. 44 miles, S.E. by E. 40 miles, S.S.W. i W. 33 miles • required the Course and Distance made good, and also what Latitude she is in. Ans. Course due South j Dist 140 miles ; the Lat. in is 14° 17' S. Ex. 3. Yesterday, at noon, we were in lat. zS' 34' N , and since then we liuve sailed N E. I E. 62 mUes, N. by E. 16 miles, E. J N. 40 miles, N.E. 3 E. 29 miles N. by W. 30 miles, and N. | W. 14 miles • what Course and Distance have w, made good, and what is our present Lat. ? Ans. Course N. 43°E. orNE. |N. ; Dist. 15!! miles; Lat. in 30° 29' N. Ejc. 4 . Yesterday, at noon, we were in lat. 44° 10' N., and since then we sailed the following courses (all true): S. 69^ W. 4 miles, S. 58° E. 15 miles, S. 66= E. 8 miles, S 66^ W. 12 miles, S. 1° E. 6 miles, S. 55° W. 2 miles, N. 21° E. 2 mUes, S 55°W. 28 miles. S. 32°E. 14 miles, S. 5 5"W. 4 miles : find what Course and Distance the ship ha» made good, and what is her present Lat. Ans. Con»gK S. I5°\V.; Disr. 55-0 miles; Lat. in 43° 17'N. 3. Current Sailing. 290. A current is named after the point towards which it runs or sets: thus, a current setting towards S.E. is called a south-east cur- rent. The mode adopted in speaking- of the wind, which is named according to tiie point from which it blows, is thus reversed in speaking of a current. f The term set, which is used to describe the direction of the cur- rent, is employed in the same way as in taking a bearing (No. 201) ; but it is necessary for tlie complete description of the current to state also its drift, that is, the distance through which the ship is carried or driven by its action. J 291. When the rate of a current per hour is known, the drift for iny number of hours is found by multiplying the rate by the number Df hours. Li like manner, when the drift in a number of hours has been * The beginner will proceed now to parallel saiUng, because, though current sailing Vl strictly a branch of plane sailing, yet some of the examples, for the convenience of arrangB- ment, involve the consideration of longitude. t It is easy to conceive that people would name a wind according to the quarter it blo»-| from, as bringing heat or cold, rain, &c., and a current according to the quaiter to which it carries them. X These terms have not in general been emjiloyed with sutticicnt precision. The term "drift" has been defined as the distance run per liour, or rate of the current. But as a seciind term for rate is superfluous, and as it is convenient to have a term expressive of the distance through which the ship has been carried by the current in any interval of time, we have used the word drift in the latter sense only. Thus the terms sel and drift are ii»;-J in speaking of the current as course and distatice are in speaking of the ship.. l.lNfJS, n.-i i.lint,^ tl.( ■ niiiiilioi- ( .riMii..s.,f.h<. Ex. by the c its rate. u;re, A ship a found to have drifted iit42 miles in 21 hours : required a.ceilaiiied, tlip ralo is fomiil by ili drift l)v tiie miinljer of Iiouin. Ex. i. A current nins z': quired its drift m 1 3 hours. '2!}2. Since tlie current sets tlie ship in a certain direction and nt a certain rate, while tlie shi]) herself is going tluough the water in another direction and at another rate, the course of a ship affected by a current becomes in general a case of traverse sailing, in which there are two courses and distances. Thus current sailing is analogous to traverse sailing, the two courses, instead of following in succession, being here considered as taking place at the same time. Tlie subjects for consideration in this section are, finding the place of a ship afl'ected by a current ; detennining the course under a par- ticular condition ; and, lastly, finding the motion of the current itself. Case I. Given the course steere<l, and (list, run by the log, with the set and rate of the current, te Hud the course and distance ii.ade good. Ex. Aship runsN.E. by N. iS miles an hour : required the Course 1 N.E. byN. 18 m. gives \V. by S. 6 m. The Course is, therefore, N. by E. { E. j Dist. 14 miles. The Construction of this example is the same as that of a case of traverse sailing, in which the courses and distances to be laid off are N.E. by N. 18 miles, and W. by S. 6 miles. 293. When a ship steering for a port is drifted by a current, it is evident that, unless it be CNactly with her or exactly against her, it will throw her out of her intended course. Since the course to be sliiipi'd in any case depends on the rate of sailing of the ship, and as this cannot be foreseen for any future hour, the course must, when it IS i)roposed to take info consideration the eHcct of the current, be detc-ruiiiu'd by the present rate of sailing, and independently of the distil nee of the port. Case II. Given the bearing of the port, and the set and rate of the current: it is required to shape the course so as to keep the port on the same bearing. 294. Bi/ Inspection, ^^'hen the bearing of the port and the set of the current are in adjocent (juarters of the compass, take their sum ; when in the same or oppusile quarters, take the difference. With this sum (or its suppleir.ent to 16 points, or 180*, if it exceeds 90°), or diHerence, as a course, and the rate of the current a« » distance, find the Dep. niles in three hours, in n id Dist. made good. current setting W. by S. D. Lat. 15 oN. D. Lat. 1-2 S. .3-8 N. Dep. looE. Dep. j;9.W WC, NAVIGATION. Will) tliis Dep. as Dep. and the rate of the ship as Dist. find (lie Course. This course being applied to tlie bearing of the port on the opposite side to that towards which the current is drifting the ship, gives tiie course to be steered. Es. 1. The port bears S. 52" W , the current sets S.S.E. two miles an hour; the present rate of saUing 7 knots : shape the course so as to keep the port on the same bearing. By Inspection. S. 51° W. and S.S.E. are in adjacevt quarters ; j^ .^y ^ ^^ the mm, therefore, of 52° and zz°4 is 74°i- 'f^'s course, with the ' ' | dist 2, gives dep. 19. The dist 7 and dep. 1-9 give the course ; t6°. This 16' applied to the right (because, hi facing tnward" S. 52° W. S.S.E. lies to the Irft), gives the Course 5 «s. 68° \V. 295. By ConstructioJi- Take a point B, any wliere, and from it lay off the s(!t and rate of the current, as BC, S.S.E. two miles; throu<j,h C draw a line A P, S. 52° VV., for the direction of the port ; from B lay off B .A., 7, tiie rate of sailins;, meeting PA in A ; then C A B is the angle Iti", which the ship is to steer to the right of the port. It is evident, in the present case, that while the ship is runn.ng along A B, looking to windward of the port, the current is setting, iier to the left towards the proposed line, A P. Attention to this point will ensure marking A on the proper side of B C ; for if a line were drawn from B towards a point between C and P, to represent the ship's course, it is evident that while on it she would be looking to leeward of the port, while the current was also drifting her to leeward. This example will serve for all cases. Thus, while the port bears as above, suppose the current sets N.N.W. 2 miles; then the point B and the line A B would lie on the S.E. side of A P instead of tiie N.W. side, the angle A would be 16° as before, but the distance A C made good by the ship in the direction of the port, would be different. EU. 2. The port bears N. 42° W., the current runs south 3 knots; rate of sailing, 5 : tfcape the Course as required by the condition. By Inspection. South giving no angle, the first course is 42° at once, which, with Dist. 3, gives Dep 2. The Dist 1; and Dep. 2 give CoimsK 24°, to be applied to the right, because in facing towards N. 42° W., south is to the left. Ex. 3. The port bears E., the curreut sets S W. by S. 3 knots ; rate of sailing, 4. East is 8 points, or 90°, which is one of the opposite quarters to S.W. ; the diflf. of S points and 3 points, or 5 points as Course, and Dkst. 3, give Dep. 2-5. The Dep. 2-5, and Di.'^t 4. give Course 39°, which, applied to the left of E., gives the Course to be steered N. 51- E. Ex. 4. The port bears S. 82° E., the current sets N. 5° W. 4 knots ; rate of sailing, j. S.E and N.W. being opposite quarters, the diff of 82° and 5^, or 77°, is the Course; which, ivith the Dist. 4, gives Dep. 39. This Dep. 3-9 being greater than the Dist. 2 Cth« (hj'i's ratoj) whii-h is impossible, slK'ns that thi' ship cannot maint.iin Ihc bcaiiiig of the port. Tiir, s\ii,iNt,s. 1 I 7 2'JC: Wln-n tlio ciirrciit sets at li^^til luigles across t)ie lino (if di- rettion of tlie port, the sliijj's velocity must evidently be e<jual, at least, to that of the current, that she may be able to stem it, and to preserve both the bearing and distance of the port unchanfred. Hence, if tlie current tend in any degree to set the ship awuy fioui her port, she will not be able to preserve the required position iiiih.ss her velocity exceed that of the current. Casein. Given the Course and Distance run by account from a ueil-determinefl place, and the true position of the ship, to find the Current. "297. Bi/ Inspection. Having the D. Lat. and Dep., both by ac- count and as deduced from observation, take the difierence between the two D. Lats. and the two Dei)s. ; if tiic D. Lats. are of difi'ereiit names, take their sum, and the same of the Deps. When the true lat. of tlie ship is to tlie nortli of the account, mark the ditf. or sum of the D. Lats. N., otherwise S. ; and when tlip true longitude of the ship is to the E. of the account, mark the ditf or sum of the Deps. E., otherwise W. Find in the Traverse Table the course and distance corresponding to the said differences, as D. Lat. and Dep. these are the set and drift of the current. Ex. 1. A ship in lat. 37° N., em\a S. 57° E., 48 miles, by account, and is found to have DiaJe good 3 i'-6 D. Lat. (S.), and 447 Dep. (E.) : find the current D. Lat. by account 26*1 1 Dep. by account 40-3 Do. true 31-6 Do. true 44'7 Diff. of D. Lats. 5-5 S. I Ditf. of Deps. 44 E. The D. Lat. 55 S., and Dep. 44 E., give Course S. 39° E., Dist. Ti, the Set and Drift of the current in the time. Suppose the time eight hours and a half, then the Ratk is 08 of a mile per hour. Ex. 2. A ship from lat. 38° 20' S., and long. 31° 15' W., sails S. 40° E., 170 miles, hj account, when she is found by obsen'ation to be in lat. 40° 54'' 5 S., and long. 30' 44'-8 W. : find the current. The lat. by account is 40° 30' S. ; the long, by account, 28'' j,' W. Lat. left 38' jo' 1 Lat. in lo 54-; True D. Lat. 2 34-5 = 154-5 1 mid. lat 40° as Course, and Diat. 30-1, give Long, left 3.° 15' Long, in 3° 44--8 True D. Long. 30-2 : D. Lat. 23-0. (See No. 318.; D. Lat. bv account 130-2 Do. true I^4•5 Diff. of D. Lats. 24-3 S. Dep. by account 109-3 Do. true 230 Diff. of Deps. 86^ \V. 1). Lat. 24-3 S., and Dtp. 86-3 W. give C. ivr of Ihe rarrent in the given lime ourse S. 74° W., Dist. 90 miles, the Set Ex. .";. (By bearings and dist. of land.) A ship at sunset sets a point of land, N. 58" E., 1 1 miles. Next morning having, as supposed, made goml S. 40 ' E. 14 miles, the point titars N. -11' E. 20 miles : required the current. The Bearing at sunset, considered as a Course from the land or S. 58° VV.. Dist 11, and S.4o"E. 14, give whole D. Lat. by account, between the ship and the point, 16-5 S. and Dep. r 3 W. The Bearing and Dist. in the morning give the D. Lat. 4-8 S., and Dep. ig-^ W. D. Lat. by account 16-5 | Dep. by account 0-3 Do. true 4-S Do. true 19-4 11-7 I 19-1 The D. Lat. u--. and Dep. 19-1, give Course or Sf.t 58° and Dist. or Diiirr 22 ; th» set it eviilenily 'from il., two bearings) between N. and W. 118 N.WKiATlON. Tlxe compleie construction of this last ease, in which longit ido is involved, requires the use of Mercator's Chart. No further direc- tions are, however, necessary than to lay off tlie place of the shij) hy D.R. and her true jwsition ; the line joining these two points shews tlic set of the current, and its drift. •298. The last example leads to the remark that, unless the ship's head be the same way at the taking of each beaiing, as well as during the whole interval between the observations, the resulting fcet of the current will be mixed up with local deviation; and the current accordingly cannot be truly determined, unless the effect of local deviation be removed. In this subdivision* rules have been laid down for working cer- tain questions in current sailing. Other matters relative to the current, which present themselves for consideration in sh iping the course, and also in determining the current itself by experiment, are treated in the division of the work entitled " Navigating the Ship." 4 Windward Sailing, 299. In windward sailing the vessel bound to a port has a foul wind. As she is thus compelled to make more couises tliau one, the case is one of Traverse Sailing; but as the course on either tack is determined by the circumstances, the inquiry is limited to the consideration of the time at which it is proper to tack. The general principle, supposing the wind to remain unchangeil, is to near the port as much as possible from instant to instant. Now the ship nears the poi-t fastest on that tack on which she looks the best up for it; if, therefore, she looks up for the port better on her present tack than she would on the other, she should stand on ; if not, she should go about. Hence it follows, that the ship should constantly keep the port in the wind's eye ; but, as working up on *his line would require tiie vessel to be continually tacking, which is practically impossible, the limits within which the rule should be followed must be determined by circumstances. The advantage of working up nearly in the stream oftlie wind towards any object, whether fixed or moving, is, that the wind cannot be worse, and, therefore, every change must be for the better.f 300. The distance run, or the ground actually gone over, is the same whether the ship makes two boards or a greater number, pro- * As it is convenient occasionally thus to refer by name to the several parts into which, from the classification adopted, the contents of tliis volume are divided, it may be stated briefly that the principal portions, as the Introduction, Navigation, &c., are here termed diviftiontt, which, when necessary, are divided into chapters. The parts of a division or of a chapter, distinguished by capital letters, are termed sectiom ; the parts of a section in lars^f italics, nn/tdh-isions, and tlic furtlier division of these, in small italics with figures in brackets, subsections, the ]jrefix sub being thus applied to the smallest divisions. t The question of closing another vessel belongs to tactics, and not to our present subject, which relates solely to the place of the ship on the sea. It may not be useless, however, to notice here, that in working up to a vessel to windward, it is proper to keep as near the stream of the wind as circumstances jiermit ; because from the time that the chase has dr(M>t lo the weather beam of the chaser, the latter, however great her superioritj' of sailing, ceiscj to neiir the chase. See Naut. .Mag. 1838, .\ri. " Chasing," p. -146. TIIK SAILINCS ii;) ci.led tint no g-ouiitl or time is lost in stays: the fipplication of tlic Btiove rule, therefore, depends entirely on the probability of a chaiij;e of wind. In this subdivision \vc consider merely the general princijile o( wilino; with a foul wind. Other points involved in Shaping thf^ Course, as the combination of a currt'nt with a foul wind, the selec- tion of such a course as may, in certain cases, convert a foul wind into a fair one, the ttteets of local deviation which have been observed while sailing on diti'erent tacks, will be treated in the Chapter or Navigating the Ship, under the heads " Shaping the Course," " Erro' of the Course." 11. 1'arallel Sailing. 301. When two places lie on the same parallel of latitude, or due east and vvest of eacli other, the distance between them, esti- mated along a parallel, or E. and W. (which is all departure), i* converted into diH'erence of longitude; or, on the other hand, theii difference of longitude is converted into distance, — by the rules of Parallel Sailing. The principles of Parallel Sailing are contained in the t\\o fol- lowing propositions. 3U'2. Prop. A parallel of latitude is a circle of w Iiich the radius is proportional to the cosine of the latitude. Let EPQ be part of a meridian, p P the pole, E Q a diameter of the e<|uator, A a place whose latitude is the arc A H Take B Q equal to A E ; then B is the opposite point to A on the same parallel. Join A B crossing C P in n. ^ Suppose now a ship to move from A round the polar a.xis C P, preserving the same lat., or the angle PC A constant; then at the end of half a ^evolution she will be at B, and PCB will be equal to PCA. Then CA and (J B being equal, each being a radius, and the angles PC A, P C B, equal, and C n common to the two triangles AC?;, BC?), these are equal (No. 117). Hence A n is equal to B « ; and this holds for every point of the parallel. Hence A and B are on the circumference of a circle whose centre is n, in the line or diameter joining any two opposite |><)ints. Now A n (see tig. p. 44) is equal to the cosine of the arc A E, C E being radius; hence CE : An :: rafl. (= 1) : cos lat., which was to be j)roved. ■\m Prop. The lentrth of a ciicMlar arc is ]iro|)()rtional to its radius. Or, the Icnglh of A B : the length of ub W C .\ ; C.(. 120 NAVIGATION. C is tlie couuiioii ceiitie of tlie arcs A B, ub. Divide the angle C into any number of equal parts, as for ex. four, by tlie lines (J D, C E, OF; join the points A and D, &e. by the chords AD, D E, &o. Then tlie sides tl A, CD, &e. being equal, and the angles A C D, D C E, &c. being equal, the bases A D, D E, &c. are all equal (No. 117.) In like manner the chords ad, de, kc. "' ure all equal. Now the triangles CAD, Cad, being isosceles, and having one angle A CD common, have the remaining angles equal; they are thus equiangular, and therefore similar (148 cor.), and their sides are proportional (146) ; hence A D : « rf :*. C A : C a. We may multiply both terms of the ratio A D : a rf by anj number without altering its value (Nos. 37 and 7), whence 4 A D ; 4 « f/ :: C A : C «. Now 4 A D is the sum of the four equal chords A D, D E, &c., and 4 ad is that of the chords a d, de, &c. Hence, The sum of the equal chords of A B ; sum of the same number of equal chords of a i : : C A : C «. This proportion is evidently true, whatever be the number o( equal parts into which the angle C is divided. It would therefore hold equally for an immensely increased number of diminished chords, as for ex. of 1', or 1", or a miliiontli of 1", or infinitely less; it therefore holds of the aic itself, which we may conceive to he composed of an indetinitely great number of indefinitely small por- tions, each of which is aic or chord indifferently,* or arc A B ; arc a b w C A ; C a. (1). If A B be tlie equator, and a h a parallel, then C A : C « :; 1 : cos lat. Whence A B : « 6 : : I : cos lat. And since Diff. Long, is an arc of the equator, and an arc mea- sured parallel to it in any other latitude is called Dep., we have, D. Long. : Dep. :: i '. COS. lat., whence Dep. =D. Long, x cos. lat. ... (I) Dep. : D. Long. : : i : sec. lat., (162 (2) (4)) D. Long. = Dep. x see. lat (2) These are the equations for Parallel Sailing. (2), These equations, in logarithms, become Log Dep.^Iog. D. Long. + log. cos. lat (1) Log. D. Long.=log. Dep. +log. sec. lat. — lo ... (2) Case I. Given the distance run on a given parallel of latitude, to find the difference of longitude. 304. By Inspection. (1.) Enter the Traverse Table with the lati- tude as a course, and look in the D. Lat. column for the given distance; the Dist. against this is the Diff. Long, required. Ab, from the nature of the case, the sum of all the chords ;h it maj- approach indefinitelv near it, the arc is said to be 1? inereaied indefinitely. THE 8A1L1NOH. 121 Ex. A ihip rune 143 miles due W. in Lat. 38' 11': required the dlff. long, she maknj The lat. 38° as course, and 143 in the D. I.at. column, give the Dist. 181, or 3° i' : the UiFF. Long, required. (2.) Or employ Table 3, as directed in the Explanation of tlic Tables. 306. By Computation. To the log. sec. of the Lat. add the lojj. of the Dist. ; the sum (rejecting 10) is the log. of the Dirt'. Long. Ex. above. Liit. 38° 11' log. sec. 01046 Dist. 143 log 2;2_,53_ DiFF. Long. 181-9 log- 2-2599 ;i(Mj. Bif Construction. Draw a line AB cast and west, and lay off 143 on if; lay off the angle BAG equal to the Lat. or 38° in this case ; draw B C per- pendicular to A B, and meeting A C in C. Then AC is the Diff. Long, re- .luired. ^ Case IL Given the Diff. Long, of two places on the same parallel, to find their distance as measured along the parallel. .307. By Inspection. (1.) Enter the Traverse Table with the Lat. as course and the Diff. Long, as distance; the D. Lat. is the distance required. Ex. The diff. long, of two places in the patalie'i of 53° 10' is 12° 14': required their dU- rsncc as measured along their parallel. The lat. ^y as Course, and Dist. 734, gire in the D. Lat. column 442 nearly : the dis- tance required. (2). Or employ Tab. 4, as directed in tlie Explanation of the Table?. 308. By Computation. To the log. cos. of the Lat. add the log. of the Diff. Long. ; the sum (rejecting 10) is the log. of the distance required. Ex. above. Lat. 53° 20' log. i-ns. 9'776i D. Long. 12 14 or 734 log. 2-8657 Dist. 438-3 log. 2-6418 309. By Construction. Draw a line A B (fig. No. .306) of any length; lay off at A the angle B A (J equal to the latitude 63**; take AC equal to the Diff. Long. 734; fi'om G draw G B ))er])en- dicnlar to A B ; then A B is the Dist. required, and measures 442. 310. In parallel sailing the Distance and Departure are iden- tical. When the course is nearly, tliough not exactly, on a parallel, the distance run and the dejiarture are very nearly ecpial ; hence it is evident that parallel sailing will apply, nearly enough for common jiurposes, to cases in which the course is not exactly east or west. 311. In lats. below .'i", when the distance does not exceed 300 miles, the Dep. may at onro lie taken as the Diff. Long., as tlt« gienlpst error "ill Reiireflv exceed 1'. 122 NAVlGATIOiN 1. Middle. Latitude Sailing. 312. Tins IS a method (founded on tlie principle of ]>ariillel sailing) of converting the Departure into Difference of Longitude, nad tlie Difference of Longitude into Departure, wlien the sliip's ffOiirse lies obliquely across the meridian ; that is when, besides Departure, she makes Difference of Latitude. Suppose a ship make 100 miles dejiarture in going, on the same sourse, from lat. 38° to lat. 41°; this departure, if made good alto- gether in lat. 38°, would give 127 Ditf. Long, by No. 304 ; and again, if made good in Jat. 41°, it would give 132-5 J)ifi". Long. Now, since the ship has sailed between these two parallels, and not on either of them exclusively, her real Diff. Long, umst be between 127 and 132'5; and therefore we may conclude it to be not far from that which would result from a departure made good altogether in the middle parallel ; hence the name of the sailing. 313. Middle latitude sailing has thus the same two cases as parallel sai'ing; and, accordingly, the rules for inspection, compu- tation, and construction, already given, Nos. 304, &c., apply equally to this sailing, observing merely to read middle latitude for latitude. 314. When the latitudes of the two places are of the same name, the middle lat. is half their sum.* In using the Traverse Tables, it is enough to take the latitudes to the nearest degree. Ej. 1. A ship sails from lat. 51° 33' N. to 49° 9' N. : find the Mid. Lat. Lat. left 52° Lat. in 49 Ex. 2. A ship sails from lat. 2° N. to lat. 1° S. The ship moving near the equator, the consideration of middle latitude is omitted, and the Dep. taken as the Ditf. Long. Min. Lat. When the latitudes are oi contrary names, no sensible error can arise from taking the Dep. itself, made good from day to day, as the Ditf. Long. But in greater distances between places in opposite •atitiides it is proper to convert the Dep. made good in N. hit. into Diff. Long, by means of the north mid. lat., that is, half the N. lati- tude, and that made good in 8. lat. by half the S. lat. When, on the other hand, the Diff. Long, is to be converted into Dep., this rule does not apply. It will be near enough for common purposes, when the latitudes are either very nearly equal or very unequal, to employ, as the mid. lat., half the greater latitude. In * The rule which directs half the difference of the latitudes of two places on opposite Bides of the equator to be employed as their middle latitude, is erroneous. The error will be readily perceived in considering a case. .Suppose a ship sails S.E. from lat. 10* N. tor 10° S ; it is evident that her ditf. long, will be exactly the same as if, on reacliing the equa- tor, she returned to the same N. lat., steering N.E., since her course is the same, and she moves in the same lats. in both cases. Thus the mid. lat., which is the average of all tlie latitudes passed through, or the half sum of the first and last, and is here 5°, is independent of the distinctions of N. and S. Tlie common rule gives o for the mid. lat. ; wlience it would follow that the diff. long, made good by a ship in ranging tlirough all the latitude* between 10° N. and 10° S., or any other equal latitudes, however great, would be the same »K if she made good her departure altogether on the equator — a conclusion manifestly erro- inK, SAILINGS. 123 an intermediate case we may combine the two mid. lat.s., <;i\ing the e;reater weight lo tliat whicli corresponds to tlie greater latitude. Ex. I. Find the mid. lat. between 30° N. and 29° S. The lats. being nearly equal, half of 30°, or i ^^, may be taken as the .Mio. Lat Ex. 2. Find the mid. lat. between 30° N. and 2' S. Half of 30^, or is", may be taken as the Mm. Lat. Ex. 3. Find the mid. lat. between 30° N. and I5°S. The N. mid. lat. is 15, tlie S. mid. lat. is 7' nearly; now the mid. lat. 15 ciirrespondh to 30° of lat., and the other, or 7", to only half as much. Instead, therefore, of dividing th« §uui of the two by 2, we give to the first double the weight of the other, and divide by 3 j llius, 15+ 15 + 7, or 37 divided by 3, gives 12°, the Mid. Lat. required, nearly. I. Given the de])arture, to Hud the dift'erence of longitude. A ship from lat. 51° 9' N. sails S.W. by W. 216 miles : required her Lat. in ami Then Course 50^ and Dep. ijg'S in D. Lat. column give Dist. 279 or 4° 39', DiFF. Long, required. Casfi Ex. 1. Diff. Long, 315. £1/ Inspection. Find the D. Lat. and Dep., and the Lat. in, ]'iiid the Mid. Lat.; then, with the Mid. Lat. as Course, look for I lie Dep. in the D. Lat. column, the corresponding Dist. is the D, Long, retjuired. By Case I. of Plane Sailing, S. 5 pointg, Uist. 216, give D. Lat. 120 and Dep. I79'6 ; bence the Lat. in is 49° 9' N. Lat. left 51 9 N. I 100 ig Mid. iMt. 50° I 316. By Computation. Having found the Dep. and the Miti. Lat., add together tlie log. sec. of the Mid. Lat. and the log. of tlie Dep. ; the sum (rejecting 10) is the log. of the Diff. Long. Ex. above. Dep. i79'6 Mid. Lat. 50° 9' Mid. Lat. 50° 9' log. sec. 0-1933 Uep. 179 6 log. 2-2543 Diff. Long. 280-3 (4' 4o''3) log- ^■4476 yi7. By Construction. (Ex. 1.) Lav oft" S C .4 the Course 5 points, and take CA the Dist. 216;' draw AB perpendicular to CS. The figure is thus far complete for plane sailing, Case L Lay off" the angle BAL equal to the ]\rid. Lat. 50°, an<l A L meeting C S is the Diff". Long. 280. Ex. 2. A ship from Lat. 29° 40' N. sails E.N.E. till she makes 72 miles D. lat. : required the Dist. run ■oil Diff. Long. By Irupeclion. By No. 276, Course 6 pointi and D. I.at. 71-9 give Dep. 173-7; and 72 miles northing give lat. in 30' 52' N. ''''/. Onne 30' 'Mid. Lat.) and Dep. 173-; IjCRO required. 124 NAVIGATION \ % xs '""w. By Construction. C B A represents the flg. for plane sailing. Lay off B AL equal to the uiid. lat. 30°; and A L is the Diff. Loiif^. and measures 201. These two examples of construction ai'e sufficient for the case. E-x. 3. A ship from lat. 44.° sS'N. runs 230 miles, and makes 56 miles southing: Imd the Course and Diff. Long. By Case IV. of Plane Sailing, p. 86, the Dist. 230 and D. Lat. 56 stand together ovet the Course 76 and against the Dep. 223'2 j then 56' southing gives Lat. in 44° a' N. The Lat. left 44° and Lat. in 45° give the Mid. Lat. 44 { or 44°. Course 44° (Mid. Lat.) and Dep. 22-3 in D. Lat. column give Dist. 3 1 : hence the Diri'. Long, is 310, or f ic'. Case II. Given the latitudes and longitudes of two places, to find the departure, and thence the coarse and distance between them. Ex. Find the Course and Dist. betwe»n C. Sierra Leone, in lat. 8" 30' N., long. 13" 8 W., tnd C. St. Roque, lat. 5^28' S., long. 35" 17'W. 318. By Inspection. Find the Mid. Lat. and the Diff. Long. o( the places; open the Traverse Table at the Mid. Lat. as a course, look for the Diff. Long, in the Dist. column, and take out tlit D. Lat. : this is the De]). required. The Dep. and given Diff. Lat. between the places give the Cours( and Dist. by Case III. Plane Sailing, p. 109. 13^ iS'W. ?S -7 W. 21 59 1319 miles. ITie Mid. Lat. of 8° 30' is 4° 15', that of 5° 2S' is 2° 44', or 4° and 3° nearly. As 4"ct>i- responds to the greater lat., we may adopt it as the Mid. Lat. (Assigning the relative wcightJ Kith some further precision gives 3^40' as the Mid. Lat.) Course 4° (Mid. Lat.) and Dist. i ;a give I3i'7 in the D. Lat. col. ; this as Dep., and D. Lot. 83-8, give Course 57°4. Dist. 1570 miles. 319. By Compiitntion. Find the Diff. Long, and the Mid. Lat., to the log. cos. of the Mid. Lat. add the log. of the Diff. Long. : the gum is the log. of the Dep. Ex. above. D. Lat. 838, D. Long. 1319, Mid, Lat. 3'4o'. Mid. Lat. 3'' 40' log. cos. 9-9991 Diff. Long. C. Sierra Leon.-, lat. 8° 30' N. Lon C. St. Roque 5 28 S. D. I.nt. 13 58 Diff. Long. Or 838 mile. Or 19 Dep. 13.6 log. log. 3- 1202 31. 93 id the D. Lat. giver 1, the Course and Dist. The Dep. being now found (No. 279.) Construction. Construct the triangle for turning the Diff. Long, into Dep., as in No. 306 (reading Mid. Lat. for Lat.). Then liaving tli(.' D. Lat. and Dep. the process is compiete<l l)y drawing the figure as for Case III. of Piano Sailing, p. lO!*. 3-iO. When the :\lid. Lat. is below .3°, and Dist. under 300 niih'-*, ► (r No. 311. THE SAILINGS. 125 Examples for Exercise. K«. 1. If 1. >hip from Tvnemoutli Castle, in Lat. 55° i' N. and Long. 1° 25' W., lails S.R. by S. 196 miles : what is her present latitude and longitude ' Ans. I.,at. in 50° 55' N. ; Diff. Long. 273m. ; Long in, 3° S' E. ti. i. A ship from Cape Clear, in Lat. 51° jO' N. and Long. 9 29' W., sails S.W. 263 miles : required her Lat. and Long. Ans. Lat 48° 20' ; Diff. Long. 288-7 . whence th3 Long, in is i+° 18' W. Ex. 3. Find the Course and Distance between Tynemoulh and the Naze of Norway. Ans. Course N. 57° 4»' E. ; Distance, 33i'3 mile* Ki. ■». Required the Course and Distance from a place A, in Lat. 51° 25' N. and Long 9" »9' W., to a place B, in Lat 36° 57' N. and Long 25° 6' V. Ans. Course S. 37° 45' W. j Distance, 1098 miles, Ex. 5. Re<|Uired the Course and Distance from a place A, in Lat. 56° 12' N. and Long. 2° 36' W. to a place B, in Lat. 57° 58' N. and Long. 7° 3' E. Ans. Course N. 71° 23' E. j Distance, 332 miles. El. 6. Required the Course and Distance from A to B ; Lat. of A 53° 18' N. ; Long, of A 0° 5s' E. i Lat. of B, 57° 58' N. ; Long. B 7° 3' E. Ans. Course N. 36° 34' E. ; Distance, 349 miles. 2. Mercators Sailino. 321. This sailing is employed for exact!}- the same pnrjooses as middle latitude sailing; but it is a perfect method, which the other is not. The calculations are performed by the help of a table of Meri- dional Parts, Table 6. 322. To find the Meridional Difference of Latitude. When the latitudes are of the same name, take the difference of the meridional parts for the two latitudes; when of contrary names, take the sum. Case I. Given the course between two places, and their latitudes, to find their difference of longitude. Ex. 1. (Lats. same name.) A ship from lat. 51° 9' N. sails S.W. by W. 216 miles: required the Lat. in and DifT. Long. 323. By Inspection. Having found the Lat. in, take out tli(? meridional parts (Table G) for it, and for tlie L:it. left ; find tiie Meridional Diff. Lat. (No. .322). With the Course, and Mer. D. Lat. in the D. Lat. column, find the Dep. ; this is the Diff. Long. By Ca,se I. No. 273, the Course 5 points and Dist. 216 give D. La^. 120 and Dep. i-'^-ei tliis D. Lat. subtracted from 51 9' gives Lat. in, 49' 9' N. I>at. in 49° 9' N. Mer. parts 3396 Lat. left 51 9 31583 Mer. D Lat. "TS7 Titc Course ; points and D. Lat. 187 give Di-p. 2S0, or 4° 40' the Diff. Long 324. By Computation. Find the Lat. in, and the Mer. D. Lai. To the log. tan. of the Course add the log. of the Mer. I). Lat.; the Hum (rejecting 10) is the log. of the T>. Long. Ex. above. Lats. 49' 9' and ^i'' 9', Course 5 points. 5 points log tan. 10-I7 5I >fer. D. Lat. 187 log. 2-27 iS DoK. LoNO. :-o-8, or 4° u/-8 loif. 2-4460 126 NAVIGATION. (This is the canon (3) No. 272. It will lie suffideiitly understood by observing that, la •ie 6g. below, C M is the Mer. D. Lat., and M L the DitT. Long., and C M : M L :; rad. • tan. M C L the course. This example is sufficient for any variety of the Case I. c 325. Bij Construction. Lay otf the course M C A, S b points W. ; take CA 216 the Dist. ; draw AB perp. toCS: the fig. CAB is, tiius far, the case for plane sailinir. Now lay ort'CM the Mer. 1). Lat. 187, and draw ML parallel to A B meeting C A Siroduced : M L is the Diff. .lono;. and measures 280. This exam])le of construc- tion is sufficient for Case L Ex 2. A ship from lat. 29° 40' N. sails E.N.E. till sne makes 72 miles D. Lat. : tind her Diff. Long. fly Inspection. Course 6 points and D. Jo" 51'- Lat. left 19° 40' Lat. in 30 52 _ 1^49 Mer. D. Lat. 84 Coarse 6 points and D. Lat. 84, give Dep. 203, or 3" 23', the Diff. Long. Case n. Given the latitudes and longitudes of two places to find the course and distance between them. E.N.E. till sne makes 7 . miles D. Lat. Lat. 72 give Dist. 188 r niles: the Lat Mer. parts 1865 Ushant, lat 48° N. long. Rx. Find the Course and Distance betwe r.nd St. Michael's, lat. 37° 44' N. long. 25° 40' W. 326. By Inspection. Take oat the mer. parts for the two lats. : find the Mer. D. Lat. and the Diff. Lono^. Enter the Traverse Table with the Mer. D. Lat. as D. Lat. and the D. Long, as Dep.: this gives the Course. Then with this Course and the true D. Lat. find the Dist., which is the distance required. Ushant, lat. 48° 28' N. Mer. parts 3334 Long. 5° 3'W. St. Mich. 37 44 2448 25 40 ^\'. 10 44 Mer. D. Lat. 886 2'o~3~7 True D. Lat. 644 Diff. Long. 1237 Then 886 as D. Lat. and Dep. 123-7 give Course 54°; and D. Lat. 644 givea 1095 miles, the Dist. required. 327. By Computation (1.) For the Course. Find the Mer. Diff. Lat. and the Diff. Long. From the log. of the Diff. Long. (adiliiig 10 to the index if necessary) subtract the log. of the Mer. D. Lat. : the remainder is the log. tan. of the Course. (2.) For the Distance. Find the course; then to its log. sec. add rlie log. of the true D. Lat. : the sum is the log. of the Distance. Ex. above. M. D. Lat. 886 ; D. Long. 1237 ; true D. Lat. 644. Dili'. Long. 1237 log. 3'0924 | Course ';4° 24' log. sec. 0-23^0 Mer. D. Dat. 886 log. 2-9474 Tr. D. Lat. 644 log. 2-S089 C.ciURHE 54= 24' Uii. 01450 I Dist. iii.6 loj 5-0419 TUK SAIUN08. ]27 6'2S. By Co'isliiicliuii. Draw tlie nieridiaii C S* tliroiigli one of thfi places, say Usliaiit, and on it lay off' the Mer. D. Lat., 886 from C to M. Diaw M L perijciidicidar to C S and equal to the Diff Long. 1237 ; join C L, and S C L is tiie Course. Lay otf CJ3 the trne D. Lat. on C S, draw BA parallel to LM and C A is tlie Dist. 1100". 32[). When the lat. is below 5" and the dist. less than 300 m., see No. 33-2. E.iam]iles for Ejcetche. K«. 1. .\ sliip, in Lat. 36" 40' S. and Long. 16" lo' E., sails W. N.W. until she arrives in Lut. 33" 10' S. : find the Dirt', of Long, and also the Long, come to. Ans. Diff. Long. 620'4\V. ; whence tlie Long, come to is 6° o' 10. fci. 2. A ship from Lat. 41° 25' N. 8nd Long. 15° 6' W. sails N.E. by E. for several days, and then finds by observation she is in Lat. 46° 40' N. : find what Diff. of Long, she has made ; also find lier Long. in. Ans. Diff. Long. 536; whence her Long, in is 6° 10' W. Ki. ;5. A ship, in Lat. 42° 30' N. and Long. 58° 51' W., sails S.E. by S. 300 miles: fi'nl the Diff. Long., and also the Long. in. Aus. Diff. Long. 219 miles; Long, in 55° 12' W Ei. 4. Find the Course and Distance between Tynemouth and the Naze of Norway. Ans. Course N. 57°4o'E. ; Distance, 331-4 miles. Ei. i. Reqviired the Course and Distance between Tynemouth and Helgoland. Ans. Course S. 81° 8' E. ; Distance, 324 miles. Kx. 6. Required tlie Course and Di.«tance from Diego Ramirez, in Lat. 56" 29' S., Long. 68 43' W., and C. Lopatka. in Lat. 51° 2' N., Long. 156*46' E. Ans. Course N. 46^ 21' E. ; Distance, 9346 milei 3. Selection of Mid. Lut. or Mircntor's Sailing. [1.] Finding the Ihjf. Lovy. 330. The difference of longitnde foniul by Mid. Lat. is true <ii the equator, and very nearly true for short distances in all latitudes «?spccially when the course is nearly E. or W. In high latitudes, when the distance is great and the course oblique, the error becomes considerable; but the result may be made as accurate as we please by subdividing the distance run into small portions, and finding the Diff. Long, for each portion separately. 33L The Diff. Long, deduced by Mid. Lat. sailing is too small- an estimate of the error for places on the same side of the equator may be formed V)y the help of a few cases. Suppose the course 4 points or 45", and the D. Lat. 10" or 600 miles; then if this D. Lat. is made good in any latitude below 30° the error of the D. Long, will not exceed 2'; if inade good between the parallels of40**an(i 50' the error will be about 3'; and between 60° and 70°, aljoiit U/, or J of a degree. For smaller distances the errors will be much * The figure in the preceding page will, after the various examples given, serve suffi. ricntly well to illustrate generally the ronstruction of this case. The learner will merely abserve, that if the other place was to the norlhward of Ushant, the Mer. Diff. Lat. C M woi.ld be laid ofl' to the northvtard of C. In like m.nnner, if the ntlier )il.iie mis to tlit f.uliittnl of Usionit, the D. Long. M 1. would be laid off to tin- mulunrd, or to the right ol 128 NAVIGATION. less, and for greater distances much greater, as they vary In much more rapid proportion than the distances.* 332. It is proper to remark that when the Course is large, thai is, near seven or eight points., tlie D. Ivong. should he found by middle latitude in preference to Mercator's Sailing: because, although the latter is mathematically correct in piiiiciple, yet a small error in ihe Course may, when the Course is large, produce a considerable prror in the Difference of Longitiule. The reason of this is easily shewn. In mid. lat. s;iiliiig we convert the depnrture into D. Long. The process increases the Dep, in a ))roportion which is less than 2 to 1 in all latitudes below 60", and exceeds 3 to 1 in latitudes beyond 70*. The erj or of the Dep., increased in the same proportion, becomes thus tlie error of the D. Long. Now when the course is nearly E. or W. the Dep. is nearly the same as the distance, and an error of some degrees in the course does not affect the Dep. sensibly ; hence in this case the error of the D. Long, depends on that of the Dist. alone. But in Mercator's Sailing, on the other hand, we convert the Mer. Diff. Lat. into D. Long., and the process, when the Course is large, converts a given Mer. Diff. Lat. into a D. Long, much greatei than itself; and thus increases the error of the Mer. Diff. Lat. in the same proportion. Thus, for examjile, at the course 80° the D. Long, exceeds the Mer. Diff. Lat. in the proporticm of 6 to 1 ; at the course 85® this proportion is 1 1 to 1. Now when the course is large a slight change in it sensibly affects the D. Lat., and also the Mer. Diff. Lat., which is deduced directly from it. In high latitudes the Mer. parts vary rapidly, and the error of the D. Long, is aggravated accordingly ; hence the prece])t more especially demands attention in high latitudes. [2.] Findrng the Course or Bearing. 333. The bearing of the jmrt is tiuly deduced in low latitudes and at short distances by the method of Mid. Lats. ; but the result cannot be rendered accurate in high latitudes by subdividing the distance, which is unknown, into small portions : such cases are truly solved by Mercator's sailing. When the bearing is large, or near 90°, the method of Mid. Lats. should be preferred to Mercator. 334. The course or bearing computed by mid. lat. sailing is too preat. The error, however, in ordinary cases, will be much less than that to which the ship's course itself is liable. 33.5. The Course as I'educed by Traverse sailing, from several courses, does not afford accurately whether by Mercator's or Middle L;ititude Sailing, the Diff. Long, made good by the ship, because the * The proper mid. lat. to employ should be somewhat greater thnn the mean of the lats. A Table has been given, by Workman ("Navigation Improved," London, 1805), shewing the correclion to be added to the mean of the latitudes, in order to obtain true results. But ("or common jmrposes the usual method, of which the recommendation in practice is its {Teat convenience, would seem to be near enough, and when more precision is required tli« :vui|>lete solution by Mercator's Sailing is effected with rer>- little more labour. (See No. 331 ) Tin: SAii.iN'os. I'JD n.ff. I-uiig. luaile good on any Course Jcpemls entii-fly up )m the liitituile in wliich the sliip actually moves. Kx 1. A ship sails from Lat. 70° N. ; 1st, N.E. 400 miles tn Lat. 74" 43', then S.E. 400 miles, when she returns to the parallel of 70°, having made Dep. 556 miles, and D. Long. 31° 18'. Kx. 2. She sails 556 miles on the parallel of 70°, making D. Long. 27° 34'. Ex. 3. Starting from 70°, as above, she sails S.E. 400 miles to Lat. 65° 17', then N.E. 400 miles to 70", having made 556 miles of Dep. and D. Long. 24*^ 54'. The 1st and 3d case, reducing the two courses to one by the Traverse Table, give tlie same Course and Uist. made good as in Case 2, viz. East 5.56 miles, or Dep. 5.56 m., and D. Long. 27° 34', which is erroneous. In Case 1, this Dep. is made good in the average lat. of 72°^ ; in Case 2, in 70°; and in Case 3, in 68°. It may appear perplexing to the student that the ship should return to the same parallel, after having made y^, given Dep., and yet that her long., that is, her position in the parallel, should be different in different cases; but lie must bear in mind that the Dep. has not been made good on the parallel, exce|)t in Case 2. If he lays off a case of the kind on the globe, he will perceive clearly the nature of the question. To olitain accurately the Diff. Long, each course should there- fore be separately considered. But, in general, except in very high lats., the distances are not large enough to introduce much ei'ror oit this account. III. Great Circle Sailing. 336. When the ship sails on a rhumb line (No. 198), her track cuts all the meridians as she passes them in succession, at the same angle; and thus, while steering a course, her head is kept on the same point of the compass until she reaches her intended port. This condition, namely, keeping tlie course constant, is the most con- venient in practice, and, besides, produces in all the calculations in which the place of the shij) is coTicerued the utmost simplicity of which they are capable. But the track on the rhumb line is not the shortest distance measured directly over the snrfiice of the sphere from one place to another, or the distance " as the crow flies," except when the course is due north or south, or east or west on the equator. The shortest distance between two points on the surface of a sphere is the portion or arc which they include of the circle passing through l>oth the points and the centre of the sphere. Such a circle is called a great circle* as distinguished from other circles whose centres do • The great cirule paj^sing through two plares may he found on a globe by stretching 1 thread evenly lietween them ; or, by turning the globe about till the twn pbucs fall on the apper edge of the wooden rim, or horizon of the globe, which thus marks the lirrlr. The distance between the points may be me.isured at once by laying the thread along the cquatoi uf the globe. The ojurses are found by measuring the angles between the thread and the oieridians ; the most convenient instrument for which is the horn semicircle, or protractor, u it is also called (No. 108) In order to compare the great circle vith the rhumb line the U:ter most be projected on the gl' ^^.. K i^O NWKiA'CION. not coinci J.e with tlie centre of the 8|>here ; as, for instance, the parallels of latitude, of wliich the centres are in the axis between the centre and the pole, and which aro called smidl circles. Hence sailing on a circle of the former kind is called Great Circle Sail- ing.* On this course, and on this course alone, the ship steers for her port as if it were in sight. The three arcs joining- two points on the surface of a sphere with each other, and with a third point, and having for their eonnnon centre the centre of tlie sphere, constitute a Spherical Triangle. In the problem under consideration the two places are the two points, and the third point is the pole, and the triangle is formed by the distance between the places and their colatitudes. Some of the rules in this section may be employed accordingly in other problems of spherical trigonometry. 337. Great Circle Sailing is adapted principally to the second only of the two cases, No. 270, or Shaping the Course; because tlie ship, even when moving on a great circle, must necessarily be kept on the same course (that is, on a rhumb line) for a short distance at n time, and her place may then be deduced by the rules already given in the preceding section with incomparably greater con- venience than it could by any rule in which the distance made good was rigorously considered as described on a circle. Although this sailing is thus restricted to one case, we shall, for the sake of clear- ness, divide the problem of finding the course by Inspection into two cases, namely. Case I. in which the places are on the same side of the equator, and Case II. in which they are on opposite sides. Case I. Bi/ Inspection. (The ulaces on the same side of tlie equator.) (1.) For the Dist. With the two lats. enter the Spherica. Traverse Table (Table o), and take out M and N. With the complement of the Diff. Long, as a Course and Dist. 1(10 (Table 2), find the Dep., and write it under N. Wl>^n the Diff. Long, is less than 90°, add this Dep. to N. ; when the Diff. Long, is greater than 90°, take the diff. of the Dep and N. . „ , .i With this sum (or diff.) as D Lat. and M as Dist. find the arc in Table 2: this is the Distance required in degrees of 60 miles each. (2.) For the Course. Having found the Distance. With the lat in, and the compl. of the Dist. in degrees, find M. and N ^W^ith the lat. to as Course and M as Dist. (Table 2), find the Dep., and write it under N. When the Diff. Long, is less than 90°, take the diff. between this Dep. and N. When the Diff. Long, exceeds 90°j take the sum of the Dep. and N. With this diff. (or sum) as D. Lat. and Dist. 100 (Table 2), find the Course. ling, lor 3 like na^ THE HAll.lNOS. ini The (JoiiJNe IS to be reckoned aceordiiip; to the following rule: Dirt. Im 11.811 90" (or 5400 miles). Dist. greater than 90° (or 5400 miles. | Dep. few than N. Course to be reckoned in N. lat. from S. in S. lat. from N. Dep. greater than N. Course to be reckoned in N. lat. from N. in S. lat. from S. , Course to be reckoned in N. lat. from N. in S. lat. from S. Ex.1. Find the Distance between St. Helena, in lat. i5°55'S., long. 5 44' W.. and pe Horn, in lat. 55' 59' S. and long. 67° 16' W., and the Course from each place to the The D. Long, between 5° 44' W. and 67° |6' W. is 61° 32' ; eompl. 28°. For the Distance. 16° and 56° (the lats.) give M i86-o N 42-5 28" (co-diff. long.; and Dist. 100 give Dep. 46-9 (D. Long, less than 90*.) Sum 89-4 The Dist. 1S60 and D. Lat. 89-4 give 61'^ nearly, or Dist. 3660 miles. The comp'.e- For the Course from St. Helena. For the Course from C. Horn. 16° (Lilt, m) and co-Dist. 29° | 56° (Lat. in) and co-Dist. 29° M 118-9 N 15-9 I M 204-5 N 82'i 56 (Lat. to) and Dist. 118-9 Dep. 98-6 16 (Lat. to) and Dist. 204-5 Dfp - 56-? (D. Long. /ew than 90°.) Diff. S2-7 Diff. 25-9 Dist. 100 and D. Lat. 82-7 give 34°, [ Dist. 100 and D. Lat. 25-9 give 75°, wliich is S. 34°W'., the Course required, which is N. 75° E., the Coursk required, because the Dist. is less than 90°, the Dep. because the Dist. is less than 90°, the Dep. t/reater than N, and the Lat. is south. ; less than N, and the Lat. is south. By Mercator's Sailing the Course is 50" from either place to the other, and the Distance J74.0 miles. Ex. 2. Find the Distance between Madeira, in lat. 32° 38' N., long. 16° 55' \V., and Bermuda, in lat. 32° 20' N.. long. 64° 51' W., and the Course from Madeira. Tlie D. Long, is 47° 56' ; the compl. 41°. For the Distance. 32° and 33° M 140-6 N 40-6 42 (co-D. Long.) and 100 Dep. 66-9 Sum 107-5 Dist. 141 and D. Lat. 107-5 give 40'', or 1400 miles, the Dist. required. For the Course. 33"^ (Mad.) and to- Dist. 50° M .S5-5 N 7,-4 32° (Berm.) and 185-5 "ep. 98-3 Diir. 20^ Dist. 100 and D. Lat. 20-9 give 78^, which is N. 78°W., the Course reiiuired, because the Dist. is lesi, than 90, the Dfp. greater than N, and the Lat. north. Kx. 3. Find the DiHrancp between a point in long. iSo° on the equator, and another in Ul. . 0° N., long. 140^ W., und the Courses between these points. For the DisUnce. Lats o* and 40° give M 130-5 and N o. Then 50° (the co-D. Long.) tnd Dist. 100 give Dep. 76-6 ; the sum of N and this is 76-6, and Dist. 130-5 with D. Lat. 76-6 givej 54°, or Dist. 3240 miles. For the Course from Lat. 0°. 0° and the co-Dist. 36' give M 113 "6, N oj 40° and 124 Ci»e Dep. 79-71 Dist. ico and D. Lat. 79*7 give 37°, which is N. 37° E., the Coursk required. For the Coarse from Lat. 40°. 40° and 36° give M i6r4, N 6i-o; o and Dist. i«i give Dep. ; Dist. 100 and D. Lat. 6i-o give 51°. which ig S. ^i°W., the Course required M the Dep. c II Ipm tli.in N. 132 NAVKJATION. 338. Case II. By Inspection. (Tlic places on opposite sides of the equator.) (1.) For the Distance. With the two lats. take out M and N. :Table 5.) With the complement of tlie D. Long, as Course (Table 2), and Disl. 100, find the Dep. When the D. Long, is less than 90°, take the difference between this Dep. and N ; when the D. Long, is greater t\mn 90°, take the With this diff. or sum as D. Lat. and M as Dist. find the Course or arc in Table 2. When the D. Long, is less than 90°. If the Dep. is greater than N, this arc is the Dist. required ; if the Dep. is less than N, take the supplement. When tlie D. Long, is greater than 90°, take the supplement of Hie arc. (2.) For the Course. Having found the Distance, with the Lat. in and the complement of tlie Dist. to 90° find M and N. With the Lat. to as course and M as Dist. (Table 2), find the Uep. When the D. Long, is less than 90°, take the sum^ of this Dep. and N ; when the D. Long, is greater than 90°, take the difference. With this sum or diff. as D. Lat. and Dist. 100 (Table 2), find the Course, which is to be reckoned as follows: — Dist. less than 90° (or 5400 miles.) Dist. greater than 90° (or 5400 miles ) Course lo be reclconed in N. lat. from S. in S. lat. from K. Dep. leas than N. | Dep. greater than N. Course to be reckoned Course to be reckoned in N. lat. from N. in N. lat. from S. in S. lat. from S. in S. lat. from N. Ex.1. Find the Distance between C. Palmas, in lat. 4° 22' N. long. 7°44'W., and Frio, in lat. 23°o'S. long. 41° 57' W., and the Course from each place to the other. The D. Long, is 34° 13' ; the complement is 65°, For the Distance. 4" and 23° (lats.) Rive M 108-9 J^' 3 >• 56° (co-Diff. Long.) and 100 Dep. 82-9 (0. Long tos than 90".) Diff. 799 Dist. 1C9 and D. Lat. 79-9 give 43°, or Dist. 2580 miles; the corapl. is 47". For the Course from C. Frio. 23° (C. Frio) and 47" M i i;9-3, N 45-5 4i (C. Pal.) and 159 Dep. 12 5 Sum 58-0 For the Course from C. Palmas. 4' {C. Pal.) and 47" M i47'o, N 7-; zy (C. Frio) and 147 Dep. 57-4 {D. Long, less than 90°.) Sum 64-9 Dist. 100 and D. Lat. 649 give 49^, which is S. 49'^W., the Course required, because the Dist. is leis than 90^ and the , bee Ex.2. Find the roursee and Diiitance between Diego Ramirez, in lat. 56^ 29' S. lona 68"43'W., and C. Lojiatkn. in lat sTi'N, long. ts6''46'E. Thr D. I^ng. is i34''-ii. thB cn-D. Long. 4;". Dist. 100 and D. L.it. 580 give cf, which is N. 55° E., tlie Coursf required, tlie Lat. is south. IJIK SAIMNOS, 1":] For tlir Distance. {<" .mil 56 j' give M iSg'o, N 1866. Then 44.^° and nisi, ico ir\f Di'p. 701 ; tin: Sinn of N. and Dep., or 2567 aa D. l.at., and Dist. 288, give 27 \ oi- Lli>r 153", or 9180 miles: the co-dist. is b-^'. For the Course from Diego Ramirez. 565" and 65° give M 399'i, N 296 6; 51 ' aiij J99 give Dep. 3100; the rfi/f. i3'4 and Dist. 100 give 82°: Coursh, N. 82° W. For the Course from C. Lopatka. 51' and 63^ give M 3500, N 242-4; 564' and 350 give Dep. 291-8; the dif. 494 and Dist. ico give 60^: Course, S. 60" E. 339. To find the Courses and tlie Distance between the places hy CompiitatioH. Find the co-latitudes of the places. If the places are on different sides of the equator, add 90° to tlie latitude of one of tiiem for its eo-hititiule. Find the D. Long., and take half of it. (1.) For the Cour.ses. Take half the sum of tlie colats. and half their diff. Add together the log. cot. of half the D. Long., tho log. sec. of the half sum, and the log. cos. of the half diffei'ence : the sum (rejecting tens) is the log. tang, of half the sum of the two courses. When the half sum of the colats. exceeds 90°, take the snpplt- nient of the resulting arc for the half sum required. To the same log. cot. add the log. cosec. of half the sum of the colats., and the log. sine of half their diff.; the sum (rejecting tens) is the log. tan. of half the difference of the two courses. The sum of the half sum ami half diff. of tlie two courses is the course from the place in the /smaller of the two co-latitudes to the other; the difference of the said half sum and half diff. is the other course. The course is to he reckoned from the N. point in north latitude, Bnrl from the S. ])oint in south latitude. Ex. I. Find the Courses on the grem circle, between St. Helfnii.iii hit. 15° 55' S., long. 5° 44' W., and C. Horn, in lat. 55° 59' S., long. 67° 16' W. The D Long, is 61° 32' ; half D. 30° 46'. Colat. 34° 1' (C. HonO 30° 46' 001.02252 02252 C.h.t. 74^,5 O^t- Heh-n.a) Slim 108 6 half sum 54 3 sec o 2313 coKec. O0918 cos. 97687 Uiir. 40 4 hull' ililf. 20 2 (•08.99729 sin. 9 5347 sin. 9 70S9 69° 35' tan. 04294 35" 24' t,an. 9-85T7" 69° 35' sec. 10^574 35 24 30 34 COS. 9 9350 CoCKSR, S. 104 59 E. from C. Horn, or N. 75° 1' E. 2 CotiKSE, S. 34 1 1 W. from St. Helena. bl 8 = 3668 ni.* Kx 2. Finil the Courses on the great circle between Diego Ramirez, iu lat. 56° 29' S., !■ n^'. 68° 43' W., and C. LopitkH, in lat. 51° 2' N., long. 156° 46' E. TheD. Long, is 134° 31'; theco-lats. 33° 31' and 141° 2'. The half sum of the required courses is 79° S', and the half diff. 18° 42'. The sum of these is the Couksr from eolat. 3^° 31'. or Diego Ramirez, S 97° 50' \V., or N. 82° 10' W.; the diff. is the CouRSK from C. I.'-i.alka, or S. 60° 20' E. (2.) P'or the Distance. Fyib )ve method,* or take the supplement of the Diff. Long, to 12'' or 180". Add together the two co-lats. Add together the log. sine square of the said supplement, and the log sines of the co-latitudes : the sum (rejecting tens) is the log. sine square of an auxiliary arc aif Write X under the sum of the colats., and take the sum and difference, and the half sum and half difference. Add together the log. sines of the last two terms : the sum (rejecting tens) is the log. sine square of the Distance required. f Log bine tcjuaro is identical with the log. Imver.sine of Inmaii's tables. 134 NAVIGATION. Ex. Find the Distaaee between St. Helena, in lat. 15° 55' S., long. 5° 4/ \V„ and C. Horn, in lat. 55° 59' S. and long. 67° 16' W. Diff. Long. 61° 32 ' Suppl. 118 28 log. sin. sq. 9-868247 Colat. 34 1 log. sin. 9747749 Colat. 74 5 log. sin. 9-983022 Sum 108 6 Arc X 78 8 log. sin. sq. 9-599018 Sum 186 14 Ditr. 29 58 5 '^'™ 9i 7 'Pg- sin. 9'999357 ^ Ditf. 1459 log. sin. 9-412524 DiST. 61° 4', or 3664 miles, log sin. sq. 94118S1 The Distance by Mercator's Sailing (No. 3'27) is 3736 miles, or 72 more. 340. The course on the rhumb line,* from one of two places to the other, is exactly the opposite of the course to that place from the other; while, on the great circle, as appears from the preceding examples, these courses are very ditfereiit. The ship, wliiie on tiie rhumb line, is always changing tlie direction of lier head with respect to her port, for which she never steers exactly until it is in sight, because this track cuts all the meridians at the same angle, and the meridians themselves are not parallel to each other ; but on a great circle she steers directly for her port, while, as the angle made by her track with the meridians is perpetually varying, the direction of her head appears by the compass to be continually changing. This track, accordmgly, is the only one on which the ship nears her port by tlie whole amount of distance which she makes good fiom intitant to instant. Great circle sailing includes the case of sailing on a meridian or due N. and S., and on the equator, because the meridians and equator are great circles. 341. While sailing at the same rate on the same rhumb, the ship always changes her latitude by the same quantity ; but while sailing at the same rate on the great circle she may change her latitude, not only by unequal quantities, but in opposite directions. For exam])le, suppose the polar seas navigable, then the shortest way for the ship to go from a point in the arctic circle (or any other parallel of north latitude) to another jioint ISO** of longitude from it, and ill the same latitude, would be to cross the ]>ole ; in which case she would first steer north and then south, whereas on the rhumb line she would constantly steer east or west. 342. The track on the great circle and that on the rhumb line difter most widely from each other in high latitudes, and between places on nearly the si-.me parallels. On the other hand, when the places are on opposite sides of the equator, the great circle and rhumb line intersect each other, and the difference between them is not so coiis))icuou9. In low latitudes, and in all latitudes when the course is nearly on a meridian, the two curves nearly coincide. 343. If the arc of the great circle passing through the two jilaces [noi bring both on the same meridim or on the equatorl be )>ro- • AUo call.d the luxodromic curve. rnr. saimnhs. 135 •liutfil bcyoiid tliem, and carried round the g)oI)o, il will pass fliroiigb two points diametrically op)iosite in latitude and longitude, which we have called vertexes, each of them heiiig the highest point in latitude S. and S., passed tlirough by the circle. The vertex is 90° from the point where the great circle between the places (or produced beyond tlieni) cuts the equator. When the course shaped on the great, circle from each place is less than 90° (reckoning both courses from the nearest pole), the vertex falls between the places. At this j)oint the ship, neither in- cieasing nor diniiuisliing her latitude for a time, steers E. or W. But when the course from one of the jjjaces exceeds 90'*, the vertex of the circle falls outside the arc joining them. 344. To find the Latitude and Longitude of the Verte.i. (I.) For the Latitude. To the log. cos. of the lat. of one of tlie places add the log. sine of the course, on the great circle, from this place to the other: the sum is the log. cos. of the lat. required. (2.) For the Longitude. Add together the log. cosec. of the latitude already employed, and the log. cot. of the course already employed: tiie sum is the log. tan. of the D. Long, between the vertex and the place worked from. Kx. 1. Find the vertet of the great cinle jjassini; througli Rio de Janeiro, in lat. 12° <;5' S. long. 43^ 9' W., and the Cape of Good Hope, in lat. 34° 22' S. long. iS' 30' E. The Course from Rio is S. 63^ 12' E., tliat from the Cape S. 84° 54' W. ; each of these courses, reckoned from S., being less than 90", the vertex falls between the places Latitude. Rio, lat. 22° 55' COS.* 9-9643 (.'ourse 63 12 sin. 9-9506 Lat. 34'- 42' cos. 9-9149 Ej. 2. Find the vertex on the great circle passing through St. Helena and C. Horn. By Ex. No. .S.S9, the Course from St. Helena is S. 34" 12'W., that from C. Horn is B. 104^ 58' E. J since one of these courses exceeds 90^, the vertex falls without. Ans. Lat. 57° 17' S.; Long. 85° 10' \V. 34.5. Wlien the sliip sails on a great circle between two places on the same side of the equator, she is always in a higher latitude than if she had sailed on the rbumb line;; hence, since both tracks coincide at their extremities, there must be a point in the great circle at which its distance from the rhumb line, measured on a meridian, is greater than anywhere else ; this point we shall call the point of Maximum Separation in Latitude. When the ship crosses the equator, there are two such points, the one being to the northward of the rhumb line in north lati- tude, and the other to the southward of the rhumb line in south latitude. 346. The track of the great circle between any two points • .*s none but the logarithmic sines, cosines, &c nre employed in this work, except in ^'l>. 'J.'i4, we shall hcni.'eforih,for brevity, dispense with the abbreviation tog. in the examples. Longitude. 22" 55 cosec 0-4096 cot. 9-7034 n Long. 52" 23' tan 0-1130 Ri 43 9W. Long. 9 14 E. 13fi NAVIGATION. may be conveniently shewn, by determining the latitude of its point of intersection with each of a certain number of intervening meri- dians, the degree of exactness being increased according to the number of meridians taken. To find the latitude of the point where the great circle passinpf through two places intersects any given meridian, Find the position of the vertex (No. 344). To the log. tan. of the lat. of the vertex add the log. cos. of the difference of long, between it and the given meridian, and the sum is the log. tan. of the required latitude. Ex. Find the latitude of the point where the great circle passiDg through St. Helena and Cajie Horn intersects the meridian of 30'' W. Vertex (Kx. 2. 344) lat. 57° 17' S., lung. S5" 10' W. Laiitiide 57" 17' tan 0'922 Diff. Longitude 55 10 eus 9 756S Required Latitude 41 39 tan 9 949<3 The log. tan. of the lat. oi tlie vertex heini; corstanf, the lais. of tlie points of iui.n. section of the great circle with au^ desired uuiubir of nieridiauo may tlms be rapidly ooniputed. .347. To facihtate the practice of Great Circle Sailing, Mr. J. T. Towson in 1847 devised a method by which, using a diagram and a table, the successive courses on the great circle can be found without the labour of calculation.* The manner of projecting the track, and of measuring the distance on Mereator's chart, are described in Chajj. V. Other matters demanding consideration when it is proposed to make a voyage on a great circle, are treated in the division of tlie work appropriated to Navigating the Ship.t • Towsoii's Tables for facilitnting Great Circle Sailiug. Sold by J. D. Potter, )4o Wiiiories. London, E. t Tlie Azimuth and Star-azimuth Tables of Burjwood and Davis also facilitate Great Circle Sailing. The lat. in being taken as the Lat., the lat. of the port biund to as the Pec, and the diff. long, as the Hour-angle, gives the Azimuth, which will be the True Course. From these the Great Circle Course may be prajeeted on the Chart. Hee Burd- Wood and Davis' Azimuth Tables, published liy Potter, 145 Minories. Kx., a ship bound from Cape Ki ;ig, entrance of Yedo Bay, to San Franci.sco. Cape King, li'- 34° 54' N., long 139° 53' E. San Francitco, lat. 37"' 4S' N., long. 122=29' W. Ditf long. 97° 38', or 6'' 30°' 32". L.it. in. Lat. bound to. ^^ Hon'r.'™cio TnTe™'oiIre'e Cnttius; Mcr. of 35'" 38° &''jO™ N. 54° E. 150° K in lai 41' 4' 38 5 50 >i- 61 K. ito !•;. „ 45 45 38 S 10 N. 6S E. 170 K. ,. 4S 48 38 4 30 N 75 !■;. i^o ., 49 49 38 3 5° N. S3 E. 170 W. „ 5u 50 38 3 10 N. 91 V:. iLo W. „ 50 N'. 100 V, 150 49 38 I 50 N. 109 K. 140 \V. „ 47 47 38 I 10 N. 119 p;. i-,o W. ,. 43 43 jS >!• «i' 1'- ^"'^a" l'r..ueiscu. CHAPTKR IV. Taking Dki'artuues, I Rt a Single Beauinc; and Distance. If. Determination ov Distance. III. Methods by the Chart. 348. Determining the place of the ship with reference to a point of land, or other position of known latitude and longitude, is called Taking a Departure. TKv' position of the ship with respect to a point of land or Cither fixed and conspicuous object is defined by the direction iu which siie lies, and her distance from it. The direction or bearing of tli3 ship from the land, being the opposite i)f the hearing of the land from the ship, is fui'nislied at once by the compass, or it may he found by observation of an Astrono- mirrd Jiraring ; but the distance from the point, when it cannot he estimated or guessed with sufficient precision, must be dcihieed by means of some further observation, taken at the same time as tiie bearing, or after an interval. When a former position of the shij) herself is adopted as a point of departure, the direction (or course) and the distance are deduced from tlie rcckunini;'. I. By a Sinole Bearing and Distance. .'Un, 'I'iie object being set by the comi)ass, its distance is esti mated by tlie eye. This, which is the common method of taking departures, is iieai enough when the distance is small ; but the error or uncertainty in the estimatiDn of the distance, which, perhaps, may be stated gene- rally at one-fifth of the whole, becomes considerable when the dis- tance is great. Distances thus estimated are generally overrated. II. 1)i:ti;i{:\iinati().\ of Distance. I . /iy two lieariiiys of the same Ohject. :ir){). W'licn tlic siiip's path lies acrn~s tiu; line of direclion of the .■>b)ect the; di-tiiiicc Clin he obtained liy two hciiriiigs and ihe distance nji. by tlie sliiji in I he inliTval oniiJic lictwccn llicin 138 I^AVIGATION. Take tlie bearing of the object, and note the number of points contained between it and the ship's head. After the bearing liaa altered not less than two or three points, note the number of points in the same angle again. Note. The course and distance netweeii the positions must be those actually made yr(W. (1.) To find the distance when the lasl bearing was taken. Enter Table 7 with the first number of points at the top and the second number of points at the side ; take out the number corresponding, and multiply it by the number of miles made good by the ship : the result is the dist. in miles at the time the last bearing was taken.* Ex. The Eildj stone bore N.W. byW.; after running W. by S. 8 miles, it bore N.N.E.: rei|iiirecl its Dist. at this l;isi bearing. The number of points between N.W.by W.andW. by S. is 4; that, between N.N.E. !\nil W. by S. is II ; uiidiT 4 at the top and against II at the side staiids 072, which multiplied" by 8 (miles), gives 5-8 miies, the Dist. required. The student can easily supply a figure. (2.) To find the distance when the first bearing was taken. Enter the Table with the supplement (or difference from 16 points) of the second number of points at the top, and the supple- ment of the first number of points at the side; takeout the multi- plier, and proceed as above directed. Ex. Find the Distance of the Eddystone at the time the first bearing (or N.W. by W. above) was taken. The second number of points is 1 1, the supplement of which is 5 ; the first number is 4 p ints, the supplement of which is 12 ; then 5 at the top and 12 at the side give tlie number 0-85, wliich nuiltiplied by 8 gives 6-8 miles, the Dist. required. When the number of points between the object and the ship'.s head at either observation is 8, that is, when the bearing is at right angles to the course, the distance may be found by the Traverse Table, by entering the table with the number of points at the other observation as a course, and the distance run as D. Lat. ; the corresponding Dep. is the distance of the object when observed at 90° from the course. 351. If the time be noted when an object is 4 points on the bow, and again when it is right abeam, the distance run in the interval on the same course is evidently equal to the distance off the object when abeam. This case is called the Four-point bearin<i. It is, however, only a case of the general problem. If a shiji having a point of land or other object at any angle on the bow, proceeds steering the same course till a position is reached where the angle on the bow is doubled, the distance from the object at the last position is equal to the distance between the two positions. The case is most favourable when from the positions chosen the object is 30° before and 30° abaft the beam ; the triangle is then equilateral. * This Table was cimstrucied nt the suggestion uf Sir F. Bcaulon.aud fii>t appe.irtd b. the Naultcul Muyazine, vol. i. ^ ~ TAKING DEPAKTUKKS. ] ;;:) The error of the required distance produced hj an error in the dist. run, is a matter of simple proportion. For example, if the flist. run be -^ of itself in error, the distance required will also bo -iV of itself in error. Hence the dist. run should not be much less than the distance required. 2. % Sound. .552. An excellent mode of determining the distance is obtained by noting- the number of seconds elapsed between seeing- the Hash of u gun and hearing the report. Sound ti-jivels, in a calm, about 1 bJO feet in one second at a temperature of 66° Fahr.; hence it is easy to deduce the following approximate rule. Divide the seconds elapsed by 5, and subtract from the quotient iV of itself; the result is the Dist. in miles very nearly. V.\. The nuan of the intervals givco by 4 guns fired from C. Shilling was 14" I leciuirtd the Dist. of the ship. 5) '4-1 i-twelfthof i-S ■■! Dist. j-6 This method is capable of much precision when the gun and the ear are at the same temperature and at the same heigbt.* A mode- rate breeze in the direction of the sound causes a variation of about. 20 feet a second in the velocity ; a strong breeze more. 3. Bj/ the Altitude of lliyh Land. [1.] When the Object in sem on the Sea-Hnri:on. 353. The distance of the visible iiorizon from the spectator is e<iual to the true de])ression or dip of the eye in Table 8, increaseil by about ^^ of itself.f Thus, if the eye be twenty feet above the sea, the horizon is d'stant five miles and about half a mile more. When, therefore, the sea-liorizon is seen beyond the object, the distance of the latter is less than the depression. 354. When the summit, or any other point of known height of ail object situated beyond the sea-horizon is seen on this line, its dis- tance is at once Known ; for since the eye, the horizon, and the olject »re in the same straight line, tiie same horizon corresponds to both the height of the eye and that of the object; the distance, therefore, between tln-se two points is, by No. 20.5, the sum of the depressions {;orres|)onding to the two heights. Ex. Vx3m the mast-head, 87 feet above the sea, the Lizanl I.,ight, tlic hcii{ht of which ij 113 feet above low-water mark, is seen on the horizon : required its distaiiee. The dip (Table 8) to 87 feet is lo', that to 223 i< 16' ; the sum 26 increased by ^ of »6, X 2', is 28 miles the DiST. required. " The nnrcrtainty to which this method is liable (ihough not worth notice in navi- palinn) may, when preiision is required, be removed, in the ordinary stiitc of the atnio- sphirc, by firing a gun at each extremity of ihe line, and taking the mean of the oKscrved intervals. t In Ibis and the follov\inc rules ,'; is used insicad of ,\ (sec No. 207), because \'i \t •n eaiier divisur than 14. Tlic diQ'cicucc \s not nuitb uutice. 140 KAVIGATION. This melliod will often be useful, but from the great un- certainty of terrestrial refraction it is impossible to assign with precision the degi-ee of dependanco. [2.] When the Object ts seen above the Sta-Humon. 365. Case I. When the height of the siuumit, or other jioiiil ol high land, is known, its distance is found by means of the altitude observed above the sea-horizon with a quadrant or sextant.* 356. The Observation. Observe the altitude of the summit, and e:^timate its distance in miles. When the altitude exceeds 3° see No. -359. 357. The Compiitntion. Alt. under 3". (1.) Correct the alt. for index error (No. 496), and subtract from it j*, of the estimated distance; the remainder is tlie true alt. W'lieji the height of the eye exceeds 30 feet, add ^^ of tlic cor- responding Depression ; the sum is the true altitude. (2.) "From the true alt. subtract the true Depression to the height of the eye, Table 8 : note the remainder. To the square of the Depression corresponding to the height of the summit add the square of the remainder (which is found at once in the column headed " Square," against the remainder as a De- j)ression). Look for the sum in the column headed " Square," and take out the Depression corresponding; from this take the remainder; the result is the distance of the summit in miles.+ Ex. 1. The alt. of a hill 2000 feet high is observed 56' ; coir, for index error, —5' ; th< height of the eye, 20 feet ; estimated Dist. 8 leagues, or 24 miles : required its Distance. Deducting -^i of 24, or 2', and 3' error, leaves true alt. 5 1'. ^luare of Depr. to 2000 ft. 2304 True alt. True Depr. to 20 ft. Ditto of Rem. 46' Depr. 67' Square 4420 Rem. —46 DiST. required 21' or miles. Kx. 2. April i9tli, 1S29, Mr. Fisher observed from the jroop of H.M.S. .Spartiate, 74, the alt- of Mount Etna, 1° 26' 30"; index corr. + 1' 30" ; lieight of eye, 30 feet ; estimate. dist. 20 leagues : required its Distance. Height of Etna, 1C9C0 feet. Square of Depr. to 10900 ft. 12 A of 60'. -5' I '4 Ind. cor. + 2 ) { Alt. I 23 True Dep. to 30 ft. -6 Rem. I 17 or 77' The distance by the chart was 57 mile Ditto of Rem. 77' + 5929 Depr. 135' Square 78250 Rem. —77 Dist. required 5S' or miles. 358. When the distance is too groat for estimation, and thealtituile low, the computation must be rejjeated. Ex. Captain Beechey observed from H.M.S. Sulplnir, the Peak of Tencrifl'e clearly defined against the setting sun ; mean of 3 alts, on the arc, 19' 32"; off the arc, 19' 50"; the * In this instance, reference is necessarily made to the use of instruments which belong yrincipally to Nautical Astronomy, and are, therefore, described in that subject, Chni.. 11. t When the height of the eye exceeds 30 feet, subtract from the s\nn of the two square! ^above) the square of the corresponding Depression. F>*oui the nature of the obiA-rwaum, it U enough to work to minutes only. TAKTXr, DEPATiTT'nrS. Ill nr.n, 19 .,. ; height of tlie eye l>l!.Ulllce. Alt. 20' Depr. -4 18 feet >ei|fht of tlie Peak, 1217: feet: n<iw S<|iiaie of Depr. to 12200 ft. 1368 Ditto of Rem. i6' t 25 Rem. 16 Uepr. 118' Square 1394 — i6 DiST. required 102' or miles. tl.ing this now as an e>lima/»l band next day by the chronometer distance s to have and repeating the work, gives IC9 miles, been 1 1 5 miles. 359. When the altitude is great, or above 3", the following rule for tiie computation is preferable to No. 357 : — (1.) Correct tlie altitude for index error, subtract from it ^ of the estimated distance in miles, subtract further the true Depr. ol the eye (Table 8), and note tiie reinaiurler. When the height of the eye exceeds 30 feet, increase the le- mainder by -^V of the depression. (2.) Add the log. cos. of this lemainder to the log. cos. of the Depr. corresponding to the iieigiit of tiie mountain ; the sum (re- jectiuir 10) is the log. cos. of an are. From tliis arc take the said remainder, this leaves the Dist. of tiie summit in miles. Ex. Mr. Fisher observed the altitude of Mount Etna. 5° 15'; height of the eye, 30 feeti estimated distance, 8 leagues, or 24 miles : required its Distance. Alt. 5° 15' Etna, ht. 10900 ft. Dep. 1° 51' cos. 9-999^74 ^°*,''^ Z\\ Remainders « eo.. 9-998^55 5° 27' COS. 9-998029 Remainder 5 S D18T. 19 iiiiles. 360. Degree of Depeiulaiice. To judge of tliis, repeat tiie com- putation, using a new altitude, varied from the former by a number of minutes equal to the extent of the probable uncertainty. For example. Suppose in Ex. 1, No. 357, the altitude doubtful, or in error, 5'; repeating the work, with the altitude 46', gives the distance 23 miles, instead of 21 : hence we in'er that, supposing 5' to be in this case the utmost prohable uncertainty in the altitude, the Jistance may be depended upon to 2 miles. The greater the altitude the more accurate is the icsnlf. 361. Case II. When the height of the land is not known, tiie distance may be found while standing directly towards it, or from it, by means of two altitudes, and the distance nin in tlic interval between them. If the course is not more than two points out of the directioii of the object, the distance run may be reduced to the change of distance of the object by means of the Traverse Table. 362. The Observation. Observe the altitude. After a consider- able change in the altitude, observe a second altitude at the samp, height of the eye. Note tiie rate of sailing. Estimate the diataiicu at each observation. 363. The Computation. Find the true altitudes. No. 357. (1.1 Find from the rate of sailing the dist. run, and reduce it when ne- wwsary to the change of distance made good in the direction of the object, th:;s, — enter Tabit; 1 with the difference between the ship's 142 KAV1GATI0N-. coarse and tlit bearing of the object as a Course, and tlie Dist. run as Dist.; the corresponding D. Lat. is the change of distance required. To the lesser altitude add half the change of distance, and sub- tract the Depr. corresponding to the height of the eye ; call this the first remainder. From the greater altitude subtract the lesser alti- tude, and the change of distance; call this remainder the second remainder. Multiply the first remainder by tlie change of distance, and divide, the product by the second remainder ; the quotient is the distance in iiiik'S when t\\e greater altitude was taken. Ex.1. Observed altitude of Mount Etna. 1° 28'; estimated distance, 20 leagues. When J 8 miles nearer, observed the altitude 5° 15'; height of the eye, 30 feet: required tho bistdnce. - 28', deducting ^, Df6omUesor5',is.-23' 5° 15', deducting f^ of 22 miles or 2', ii Lesser Alt. 1° 23' 4 Dist. run +19 Depr. --_6 ist rem. i 36 or 96 Greater Alt. Lesser do. i''2 3'\ Dist. + 38 } 2d rem. or 5° 13' 1^ tl:e„''''^» ,9 miles. the Dist. required. Kx. 2. Observed the altitude of Dunnose 41', estimated distance 4 leagues or 12 milos After running -J miles directly from it observed the alt. 20'. Height of the eye, 10 feet. The 1st alt. reduced is 18'; the 2d, 40'. The Ut rem. is 18-7 ; the 2d, 14-5 : the Dist. required 9*7 miles. 364. Degree of Deperidance. This may be estimated by repealing the work with a new lesser alt., and also with a new change of dis- tance, differing from those used before by 1', and comparing these two results with the first. If they do not differ much, the case is evidently but little affected by small errors; if, on the contrary, they differ more than 1', it is shewn that errors of observation are increased in the result. Thus an error of i' in the lesser alt. produces in Ex. 1, above, only 0-3 of a mUe error In the distance required, while in Ex. 2, the latter error is t'z. Again, an error of i niilj in the change of distance produces in Ex. 1 only 0-7 of a miie in the r«sult, while in Ex. 2, it produces 24 miles. In ordinary cases an error of 1' or 2' is more likely to occur in an alt. than an error of 1 or 2 miles in the change of distance; and as precision is of less consequence in the greater than in the lesser alt. the value of the result will depend principally on the lesser altitude. The less the ist rem. is with respect to tlie 2d, tiie less is the effect produced by the above errors on the result. Thus, in Ex. 1, the 1st rem. is to the 2d, or 96 is to 192, as i to 2 nearly, and the case is good. In Ex. 2, on the contrary, the 1st rem. i8'7, is greater than the 2d, 14-5, and tlie result could not be depended upon within 2 or 3 miles. 365. Since these rules suppose the object to be referred to the eea-horizon, they apply to all cases in wliich the observer, though near the land, can descend so near the surface of the water as to tjbtjiin a perfect sea-horizon. On the other hand, whfn the latt<i in very distant, or the altitudfl TAKING DEPAHTUKKS. 143 \ery sin;ill, the methods in this section must not be too confidently .iepended upon, especially in a calm, or when, from heat, vapour, or other cause, there is anything unusual in the appearance of the horizon. Useful tables of Vertical Danfjer Angles of heights from 50 to 18,000 feet, to distances off; from one cable to 1 10 miles, have beeu calculated by Lieut. S. T. S. Lecky, R.N.R. Published by George Philip & Son. London and Liverpool, 5th Edition, 1890. III. MrnioD.s by the Ciiakt. . 1. Crn.t!! Buarings. SC6. The true bearings of two points of land being obtained, draw lines throuirh them on the chart in the directions of the bearings ; these lines cross in the place of the ship. Or a true bearing of one of the points of land may be obtained, and an angle measured by the sextant (Nos. 4S5-504) between it and a second point, when the second point cannot be con- veniently seen from the compass. 367. When the difference of bearings is near 90°, this is the most complete of all methods ; but if the difference is small, as for example, less than 10° or 20°, or near 180°, the ship's position will be uncertain, because a small error in the bearing will then cause a great error in the distance. 2. By Two Angles between Three Ohjecta. 368. When the ship's place is required to considerable accuracy, as, for example, in recovering a lost anchor, verifying the soundings on the chart, or other purposes, it should be determined bj' means of two angles observed between three objects on shore. (1.) A convenient method of laying down on the chart the angles observed, is to draw with a pencil on tracing or transparent paper, or on paper oiled for the purpose, lines containing the observed angles ; then, laying this paper on the chart, and moving it about until the lines diawn pass over the respective objects. The angular point where they meet will shew the true place of the observer. The horn protractor (No. 108) may sometimes be conveniently employed, as lines may be drawn on it with a pencil.* 3()9. By CoiiKt ruction. The observer is always on a circle passing through his own place and any two objects (No. 103) ; also the angle • The Siation-poinfer, an instrument userf in this case to fix a ship's position, cun- tisti of three (lat rulers, two movable from a common centre riglit and left of the tliirj, ichieh is fixed. The angular distance at which the mov.ible rulers are required to be (,laied on either side uf the fixid ruler being measured by an aitaclnU circular arc. 144 NAVIGATION, siibteiuled by the two objects is the same at all points of the cir- cumference on one side of the objects (No. 140). Hence, b}' ob- serving this angle and lajing it off, he can draw the circle on which he is, but cannot determine his position upon it. If now he adds a third object, he can di-awa second circle passing through this and either of the other two, and his place is the intersection of the two circles. Dbje E». 1. Lei ABC be three ihe chart ; the angle between . Tornied at O, the observer, is 46 ; that be- .■ween B and C is 30°. Join A B,B C ; lay oiT the angles BAM, A B M, each equal tp the complement of 46", i>r 44^ ; then the intersection of the lines AM, B M, is the centre of the circle ABO. In like manner lay oft' BCN, C B N, each eijual to the complement of 30^, or 60^; then N is the centre of the circle C B O, and O is th2 place of the observer. The drawing of the figure is materially simplified, in practice, by the bearing of the middle object, as this shews where the lines must fall. Ex. 2. Tlie angle between two objects A, B, is 47°, that between B and C is 107^ Lay off ABM, BAM, each equal to 43'; M is the centre of ABO. Lay off C B N, B C N, each equal to the complement of 107^, or 17' , then N is the centre of C B O. ^^■•M 370. Demonstration. Having laid ofi' two equal angles A B M BAM, and described a circle from M the point of intersection of AM,BM,bisect AB (fig. Ex. 1) in m, and join »<N; also take a point O any where in the cireumfei-ence, and join O A, O B. Then M m is ]ieip. to A B (No. 144), and also bisects the angle A M B (cor.) or A M m is half A M B. Also A B at the circum- ference is half AMB at the cer.re (No. 139); hence, AOB and rn M A are equal, and m A M tht • omplenient of A M m is also the complement of A OB. A circle therefore has been described which has the given angle at the circumference. The same proof applies when the angle at O e.\ceeds 90°. Thus, in fig. Ex. 2, BOC, icy". i« measured by half the arc B D C (supposing the circle completed, and B D, D C, joined), which is therefore 114°. Hence the arc BOC is 360° — 214°, or 146°, and the angle BDC measured by half this, is 73° ; B N C is 2 x 73°, or 146°, and N B C (or N C B its equal), which is the complement of half BNC, is 90°— 73°, or 17°, which is the complement of 107'^. 371. It is evident that the place of O is most distinctly marked when the circles cross each other at a considerable angle ; and, on the other hand, that the result is unsatisfactory when the two circles nearly coincide, or when their centres are near together. There ctinilitions govern the choice of objects. 272. In thus fixing the ship by two angles observed between TAKING DF.PARTURKS. M5 three well-linown objects on shore, tlie centre object should alway? be the nearest; for if the ship should happen to be on the circum- ference of the circle passing through the three selected points, her position cannot be obtained by the means of two angles only. A true bearing of one of the objects is therefore desirable. It will readily be seen that in war time, when the compass may be kncckcd away, or rifle-fire may make it undesirable to expose the person more than necessary, a sextant ofters great advantages, as an^ks can be obtained from any position whenie the ol.jeelg are visible. It is this contingency that makfS it especially disiiable that sailors should become expert in the method of fixing a sbip's position with the sextant. 3. By the Sounduujs. 373. When the depth of water is not great, .and al.so varies sensibly with the distance from the point of land set, this distance may be found from the chart by means of the soundings. 4. By a Bearing, and the Lat. or Long, of the Sliip. 37 i. When the lat. of the ship is known, the true bearing ot a well-fi.xed point, less than 4 points from the meridian, ornot much more, affords a very accurate departure. In like manner, when the long, of the ship is known, the bearing of a given point more than 4 jioints from the meridian, or not much less, affords the departure. In certain eates the hearing (alone) of a point of laml may be deleruiiiied from the li/iig. by chronometer. See Sumner's Method, p. 363. CHAPTER V. Charts. I. TT.«E OF MeHCWTOR's ChART. II. CONSTROCTION OF ]\IeRCATOK'S CiiAKT. III. Properties of Certain Puciectkin.s. .S75. A CHART is a map or plan of a sea or coast. It is con- structed for the purpose of ascertaining the jjosiiion of the ship with reference to the land, and of shaping a course to any place. 870. In charts, the upper part, as the spectator holds it, is the north, and that towards his right hand the east, as on the compass card ; latitude is accordingly measured between the upper and lower edges, and longitude between the right-hand and left-hand edges. Parallels of latitude and meridians are drawn at convenient di- visions of latitude and longitude. Compasses are described, by means of which a line can be re.adily drawn in any proposed direc- tion ; and the variation is marked where convenient. The depth of water, iit low water springs, is denoted, as also, in some places, the quality of the bottom. The directions and velocities of currents are expressed, and on some occasions the prevailing winds are marked.* • C'l.arlB are also constructed for special purposes, as variation charts, to exhibit iha Tariation, as well as current charts, wind charts, and ice charts. CatUion. — In purchasing A Imiralty charts care should be taken to see that they are eorrected up to date. The dates of large corrections are noted on the middle of the luwei tilge; and of imall correclioiu, in the lower kfl-hand corner of the chart. L 146 NAVIGATION. 377. Besides charts employed in general navigation, pZnTw of harbours, ports, islands, or small districts, are constructed on a ditferent scale, for reference when the ship is close in with the land. On these plans are inserted, besides the above particulars, the leading marks for channels or for avoiding certain dangeis, anchorages, places convenient for landing, and for watering, with numerous other details proper to maps. Plans of these kinds are often inserted, for convenience, in a corner of the general chart.* 378. As the surface of the globe is round, while that of the paper is flat, every chart exhibiting any extent of surface is neces- sarily an artificial construction, or, as it is called, projection, of the real state of things. The charts used in navigation are those on Mercator's projection, because on this alone the track of a ship always steering the same course appears a straight line. 379. On Mercator's Chart all the meridians are parallel, and the degrees of longitude are all equal, being the same as those of the true difference of latitude. The degrees of latitude are unequal, being extended at each latitude beyond their proper lengths, in the same proportion as the degrees of longitude on the globe are dimi- nished ; they are consequently greater as the latitude is greater. For Ex. tlie degree of lat. 6o°, that is, between 5g\° aad 60^°, is duuble of 1° ai llie equator, being increased in tlie ratio of llie sec. lat. : I. I. Use of Mercator's Chart. 1. Positions on the Chart. 380. To find the latitude and longitude of a point on the chart. Through the given point lay a ruler parallel to the nearest parallel of latitude, and look at what degree and minute the edge cuts the graduated meridian at the side, on which the latitude is wiarked. In like manner lay the ruler parallel to the nearest meii- difln, and see where the edge cuts the graduated parallel of latitude at the upper or lower edge, on which the longitude is marked. Or measure, by the compasses or otherwise, the distance of the given point from the nearest parallel of latitude, and setting off this distance from the same i^arallel on the graduated meridian at the side, note the degree and minute there expressed. In like manner, for the longitude, refer the point to the nearest meridian, along the graduated parallel at the upper or lower edge. 381. To find the bearing or course on the rhumb line between two places. Lay the edge of the ruler on the places, and refer it to the nearest compass. Or, hold the thread of the horn pn)tr;ictor (No. 108) on one of the places, and placing the centre and the zero on a meridian, slide * The p:iper on which charts are printed has to be damped. On drying distortion takes )ilaee, from the inequalities of the paper. Tliis disiortion variis jjreaily with dim-rent paper. It does nut affect navifjation ; hut aiigles taken to ililFerent points will not alw.iys agree when carefully plotted, e-p eially if the lines to the objects be long. The larger the chart, the greater the amount of this disturdon. CIIAKTS. 147 it witli tlio other liai.il ii]i or down til] tlio thread covers both the places; the bearing then will be read uti' on tiie graduated edge. 382. To find the distance on tlie rhumb line between two places. (1.) When the places arc on the same meridian. Find, by means of the ruler, where their parallels of latitude meet the graduated meridian at the side : the Oiff. Lat. they include is tlie distance. (2.) When the places are on a parallel of latitude. Take one or more divisions of the graduated meridian at the parallel in the compasses, and measure with this the distance of the places; or proceed as directed in (3). (3.) When the places lie obliquely. Take the distance between tlicm by a pair of compasses, and lay it on the graduated meridian so as to be middled by the middle parallel between the places : the D. Lat. is the distance. Of the above modes of measuring distances on the chart the first is accurate. The other two are only approximate, though near enough for common purposes. When precision is required, the 2d case, which is Case II. of Parallel Sailing, must be solved by No. 307, 308, or 309, as the chart affords no facility. In like manner, if tlie jilaces are nearly E and W., the distance should be found by Case II. of Mid. Ijat. Sailing, (). 100. In the 3d case, the construction described in No. 328 must be eni])loyed. For this the chart is particularly adapted, as it shews the Mer. D. Lat. The true D. Lat. is to be taken from the scale of longitude. 383. To lay off a point on the chart in a given lat. and long. Lay a ruler through the lat. at the side, and parallel to a parallel of lat. draw a pencil line. Do the same with the longitude. 384. The course and distance of the ship on the rhnm!) line being given from any point, to find her place on the chart. Lay the ruler through the given point, in the direction of the course. Take the given dist. in degrees and minutes from the graduated meridian, so that the parallel of lat. which the ship is upon shall middle it; lay off this distance along the edge of the ruler from ihe given point, and the ship's place is determined. :W). To lay down on the ciiart the position of the ship as given by observation. Lay off the given latitude and longitude as directed, No. 383. To lay down on the chart the position of the ship by D. R., that is, by her course and distance from a given point of departure; as, for exam])le, her place at last noon. Lay off the course and distance as directed in No. 384. Marking the ship's position on the chart is called pricking tlie fhip off. 2. Projection of the Voyage on a Great Circle, 386. The Great Circle track between any two places may be accu- rately traced on a Mercator's Chart., by determining the latitudes of its points of intersection with any desired number of inter- vening meridians. These hits, may he computed (34G), or found by the aid of Towsou's Tables or Davis's Azimuth Tables (347). 148 NAVIOATIOV. 387. But since the course and distance are liable to irregulari- ties of which the Dead Reclcotiing can take no account, a sailing ship especially' cannot be kept for any length of time upon a pre- Bcribed track ; and since, when she has once deviated from the intended line, the course must be shaped anew, it is evident that the accurate projection of a proposed voyage on a great circle sometimes would be waste of labour. It will accordingly be suffi- cient, in general, to project the track roughly. .S88. The following method by Professor Airy, for drawing on a Mercator's Chart the arc of a great circle between positions on one side of the Equator, is very simple and sufficiently accurate for practical purposes generally. 1. — Join the two points. liet>vcen which it is requ'red to project the great circle, by a stRiight line. Bisect this line, and from the point of section erect a perpendicular to the line on the side next the Equator, continuing it, if necessary, beyond the Equator. 2._Wiih tlie middle hititude (hetwecn the two places) enter the following table, and take out the " corresponding pa-aliel.'* 3.- The centre ot the arc of the great circle, required to be drawn, will be the inter- Bection of this parallel with the perpendicular. 78 16 74 S9 71 26 67 S« 6.1 37 S9 25 SS ■; ■io ^b 46 41 18 M> :?i V ss 26 42 21 42 16 39 s 9 15 !^ 14 32 "S 19 50 25 9 J3J3o 30 S 35 52 a 41 '4 1 46 37 ^, 52 . S 57 25 <2 62 51 80 N.B — If greater accuracy is required the curve of the Great Circle should he drawn by the methods of Godlray, Tow- son, or by computation. 28 30 32 34 36 38 40 42 44 46 48 50 52 " 35 S4 6 24 56 I 13 389. Godfray's Great Circle Chnrt and Course and Distance Diagram answer all the coiiditions ot" great circle sailing as complet ly and as simply as Mercrtor's Chart does for sailing on a Rhumb. The track is a straight line which may be drawn "nd examined; then the various courses and the distances to be run upon each course are obtained, as also the distance fiom the ship to her destination, by a mere inspection of the diagram.* 3. Figures of Dif event TracJcs. 390. The track of a ship by Mercator's or by Middle Latitude Sailing, appears, as before stated (No. 378), a straight line on Mer- cator's Chart, on which the meridians and parallels of latitude are represented as straight lines. But on the globe such a course, unless it be N. or S., is really a sj^rroi, winding towards one of the poles, which it can never reach. A ship's keel cannot pass over a point which is kept at any angle on the bow. ♦ See Chart Diagram for the . fcold by J. D. I'ot eilitate the practice of Gr mmation of Courses and D' 145 Minories, London, E. e Sailing, with accompanying by Hugh Godfray, Esq., M..V. CHARTS. M9 The track by Parallel Sailing, on a circle on wliicli the ship always maintains the same distance from the pule, also appears a straicrht Ime upon the chart. Tlie track by Great Circle Sailing, excej)! wlicn on a meridian, appears on Mercator's Cliurt as a curve line. It may at ttrst seem inconsistent that a curve line can, in any case, represent a siiorter distance than a straight line ; but every point of this curve lire is nearer the pole than a point in the same longitude on the track by Merciitor: and accordingly, if we divide the curve into small por- tions, and measure each portion as in No. 382 ('2), or (3), in its o«ii latitude, we shall iind that the whole distance measures absolutely less titan the length of the rhumb line joining the places.* ll. CONSTUUCTION OF MeRCATOr's ChART. 391. The following itistructioiis are merely general : practice will supply details. In N. Lat draw a line along the foot of the paper for the parallel of lowest latitude. In S. Lat. draw the line along the top. Divide this line into degrees and parts, as 30', 15', 10', or 5'. Draw at the sides two perpendiculars to this line, for tiie graduated meridians. Find, by Table 6, the Mer. D. Lat. between the lowest parallel and 1®, or 30', &c. above it. Take with the compasses this Mer. D. Lat. from the equally divided parallel, and set it off from this line on tlie meridian to be graduHted. Find, in like manner, the Mer. D. Lat. between the said parallel and 2", or 1°, ke. above it. In this way the meridians are graduated. Parallels and meridians being drawn at convenient intervals, and the points of the coasts laid down, the coast-line is filled in by hand. III. PltOPKRTIliS OF ClORTAIN PROJECTIONS. 392. Since a small portion of a globular surface may be con- sidered, in a practical sense, as a plane, charts of coasts, and maps of * In order to verify, on a globe, the results of calculations relating to the great circle and the rhumb line, the latter must be projected on the globe. To do this, note on the chart the Intitude and longitude through which the rhumb line passes, at each 4" or 5°, or less, accord- ing to the degree of precision required ; then lay off these points on the globe, in tlieir several i4ts. and longs, by means of the moveable meridian. A curve traced by hand through the p<iints laid off will represent the rhumb line nearly ennugli. If the rhumb line between any two places, difTcring cnnai<lerably in latitude and lon^- tmie, be produced on the dwirt, and Inuulerred tlius to the globe, it* spiral figure will be iia'incUy iwrccived. 150 NAVIGATION. districts of limited extent, constructed from a scale of cqnai parts, exhijjit, iike the ulaii of a building or an estate, the relative direc- tions and distances of the places upon them very nearly. On this projection, divisions of latitude and longitude may be laid off in their due proportions by means of parallel and jjerpendicular lines, drawn at jiroper distances. In drawing these lines the minute or jnile of latitude is taken as the unit of measure (Nos. 186, 199), and the ])ara]Iels of latitude drawn through certain divisions. The length of a minute of longitude being to that of a minute of latitude as the cosine of the latitude to the radius, is determined by No. 304, 305, or 30(). On a small portion of tlie surface the minutes of longitude are nearly ecpial, and the meridians are therefore drawn parallel; but if the extent of latitude be increased, the meridians will cou- vorge sensibly towards the polar side of the chart (No. 191, note *) and the character of the projection changes.* 393. On Mercator's Chart the figure of each small district or portion o*" surface is truly represented, as in No. 392 above; but, as the mile or minute of latitude, which is the unit of measure, is of a dift'erent magnitude in every different latitude, if we take a greater extent of latitude we introduce a new scale of measurement. A small island, for example, near the pole, is represented, in regard to its shape, as truly as another near the equator, but on a larger scale: Jionce, though each small portion is truly figured, portions in differ- ent latitudes cannot be directly compared. The appearance of dis- tortion of the countries on Mercator's Chart arises, therefore, from the distances in each latitude being drawn to a different scale. This projection represents, with perfect accuracy, the relative positions of places as respects a rhumb line ; it does not, however, exhibit the relative distances between places, which, when required with precision, must be found by the proper construction. No. 328. 'I'he projections here described become identical at the equator. 394. Every bearing, obtained either by means of the magnetic n-jeiUe or astronomical observation, is a horizontal angle on the surface of the sphere, formed at the eye, and contained between the meridian of the observer and a line drawn from the eye to meet a plumb-line passing through the point set. Such angle is the same thing as the course on a great circle. Hence observed bearings are never, unless due N. or S., or E. and W. on the equator, identical with bearings taken from Mercator's Chart. The diffeience is not, indeed, perceptible on common occasions, on account of the small- ness of the portion of the sphere within the view of the spectator ; but in charts of high latitudes, graduated with n)uch precision, it becon)es manifest, and must be taken into consideration when it is * In the Plane Chart the degrees of latitude and longitude are all made equal. Tlu» orojection represents very nearly the relative directions and distances of places iica.- the Jquator, and serves for plans of ports and seas in those regions; but in higher latiuiiits it bxhiliits truly no directions but E. and W., N. and S., and no distances but tliosc on a m ridian. Henee tlie figure of every portion of surface, however smuU, is distorted. Thi»e ihartsar* no lunijer used. SOUNDING. 151 r««|iiiie(] to cm|)Ioy the observed bearing oi" a distant rnoiintain for any [lurposo in whicii precision is necossary.* A distant oi)ject cannot, accordingly, be correctly laid down on tlio chart, from its observed bearing an(i distance, except in low lati- tudes; it must therefore be laid down in lat. and long, as determined by Splierieal Trigonometry. The line drawn from the observer's ()laeo to this position laid down is then the bearing on the chart, — not the direction of the object, but the course wliich a ship must preserve in approaching it while crossing all the meridians at the same angle. It follows, in like manner, that three objects which lie in the s.iuie great circle (not the merid. or the equator), and therefore, when seen in a certain direction, appear in one, form, on the chart, an elongated triangle, the middle object of the three being on the polar side of tiie line joining the extremes. Tims the summit of Mount Athos, which lies a little (0' 39") to the N. of the great circle passing through Mount Olympus and the summit of Imbros, appears, on the chart of the Archipelago, nearly 2' to the N. of the straight line joining the two latter places. ',\\)o. Tiie bearing of a distant object, as taken from the cliart or computed by Mercator's or Mid. Lat. Sailing, may be converted, ap|)roximatfly, into tlie true azimuth, as it would be observed, thus: — Find half the Diff. Long, between the pl.ace of observation and the object, and also the Mid. Lat. between them. To the log. sine of half the D. Long, add the log. sine of the Mid. Lat. ; tiie sum is tlie log. sine of the corr. required. Apply the corr. to the N. in N. Lat , and to the S. in S. Lat. • Ex. The observer in N. lat. 40° 2' sees a peak in lat. 40° 9' N., and 1° 5i'W. of hliu 1 rei)uircil the true azimuth, as deduced from the rliunib course ? The Course by Mercator's Sailing, is N. 85"i6' W. n. Loiig. 114', half do. 57' sin. 8-2196 I Rhumb bearing !!5" 26' Mid. Lat. 40° 5' sin. 9-8088 Sub. t,j Corr. 37' sin. 8-0284 I Tkiik Azim. 84 49 CHAPTER VI. Sounding. 39fi. Soi'NDiNG is ascertaining the depth of the water. This is commonly done by a lead attached to a line marked at certain divisions. • Tliia point, and also some considerations relative to the projcition of the gri'at circlei on MircMtor's Chart by rei-taognliir co-ordinates, are treated in the " Trailu d- Gi'odesie i rU^ige dci* Marin.s," par 1'. Begat Paris, 183'.). 152 SAYUiATION. .'31)/. The soundings marked on tlie cliart are taken at low-water «|iiiiig-tides; the depth is noted in fatlioms, and, in small depths, in fi-et, and the nature of the bottom is specified. The " low water" of tiie charts is, general!)', the average of the spring low water.* Since the ship's place on the chart can thus be determined, witliin certain limits, by the soundings, it is always a proper precautio-i, however correctly the reckoning may be kept, to sound on approach. ing the land. In like manner, in a fog or during the night, the navigation is often made to depend upon the lead alone. 398. Two leads are employed for sounding, the hand-lead weigh- ing 141bs. and attached to about 25 fathoms of line, and the deep-sea lead, weighing 281bs. and attached to 100 fathoms or more of line wound on a reel. A small lead of five or six pounds is sometime." used. The quality of the bottom is ascertained by fixing a lump ol tallow, called the arming, on the lower end of the lead before it ie thrown into the sea. 399. In using the hand-lead, the leadsman, standing at the ves- sel's side, or in the channels, throws the lead as far forward as he can, swinging it once or even twice over his head to give it increased force, and endeavours to draw the line tight from the lead at the instant the ship by her progress places him jierpendicularly over it. The hand-lead descends about 10 fatlioms in the first six seconds, according to some trials made by Ca])t. Bullock ; hence, when the vessel is going fast, it is often difficult to get soundings. The line is marked at 3, 5, 7, 10, 13, 15, 17, and 20 fathoms.-^- These depths are called marlts, and the intermediate ones deeps ; for exauiple, in obtaining 10 fathoms the leadsman cries, with a peculiar song," By the mark ten;" in 9 fathoms he cries, " By the deep nine." On some occasions the leadsman describes the bottom as hard or soft. The only fractions of a fathom used are a half and a quarter; ilms, 7^ fathoms are called, "And a half seven;" 7j fathoms are called, " A quarter less eight." 400. In heaving the deep-sea lead, the lead is carried to the fore part of the ship, as the weather cathead or fore-chains, or the lee cathead, if the ship is making much leeway, the line being passed along outside. The ship's way being reduced when necessary, the lead is dropped and the soundings are observed by an experienced seaman at the quarter. The deep-sea line is marked at each 10 fathoms by the corres]ionding number of knots, and with a single knot at each five. The error of the soundings is generally in excess, because the line can rarely be stretched straight fiom the lead. 401 In sounding in deep water in small vessels, whicii drift to Icewaid rapidly upon losing their way, it is generally advisable to drop the lead before the headway ceases, and to cause the vessel to * As this average height is not indicated by nature, the seaman should bear iu mind (bat tlie water may, under the influenee of strong winds, fall quite a foot below tins Bveragi'. t These divisions rciiuirc to he measured or rtclified fromliuje to time ; when ihis i« doije, the line thould be thoruuglily wetied. CHARTS. 153 gather steinway so as to pass over the lead, which will thus have desceiuled through a considerable depth perpendicularly. 4U2. The interruption to the voyage, and the inconvenience of rounding the ship in order to allow time for the deep-sea load to descend to the bottom, have led to the invention of instruments for sounding without stopping the ship's way.* Burt's buoy and nipper is a simple and well-known instru- ment. The line being rove through a spring-catch in the buoy, the lead is hove, and the buoy afterwards drojiped into the water; tho line then continues to run through the catch till the lead reaches the bottom, or is checked by a pull, when the catch firmly seizes the line, attaching the buoy to it at the depth descended through by the lead. Massey's machine registers the depth by wheelwork set in motion by a fly. — Erlccson's machine measures the depth by the space into which the contained air is compressed. Sir W. Thomson's Sounding Machine consists of a drum on which is wound about three hundred fathoms of steel piano- forte wire. This is kept at intervals between the casts in a box hlled with lime water, which entirely protects the wire from rust. A brake, partially self-acting, is arranged by a cord round a groove in the circumference of the drum, with two weights attached, one of lead (3 lbs.), the other a long iron weight (-56 lbs.). When ready to take a sounding, the brake is released by holding up the heavy weight and allowing the small one to hang freely in a recess in the heavy one. This opposes a slight resist- ance to the wire when running out, and when the sinker reaches the bottom the brake is put on by easing down the heavy weight gradually until it is supported by the small one. Eetween the sinker (which is of iron, with a hollow at the bottom to receive the arming of tallow) and the depth gauge there is a two-fathom length of plaited rope, and the same between the depth gauge and the wire. It is important that -plaited rope should be used, not twisted. The dejith gauge consists of a brass case about 2 feet long, containing a glass tube coated inside with a chenjical preparation; this tube is open at one end, and is placed in the brass case with the open end downwards. As the sinker descends, the increased pressure drives the water up the glass tube, and the height is registered by the mark made by the combination of the water .'jiid the coating of the tube ; this mai-k, when applied to the graduated boxwood scale, shows at once the depth that has been reached. There is also a counter attached to the wheel that shows approximately the number of fathoms of wure run out. The instructions sent with the apparatus are ample, and the use of this simple machine is easily learnt ; but men should be drilled at it in fine weather, so as to be able to handle it readily in bad. An officer and two men can with ease take soundings in 100 fathoms every quarter of an hoin- from a vessel going at any oi dinary speed. * IJiicmly an in>.: lumcnt has Kcii iutiuducid wherein the dcjMh is indUatc'l *>] hjdiu:>ial c jire<fiui«. 154 NAVIGATION. CHAPTER VII. The Ship's Journal. I, Keeping the Ship's Journal. II. The Day's Wokk- I. Keeping the Ship's Journal. 403. As the keeping of the log or journal, in the Royal Navy and in the merchant service, is a matter strictly professional, and as no one would be intrusted with it whose experience did not qualify him to know what matters to insert and how to express them, — and, moreover, as the log- board, from which the ship's log is copied, is ruled in an established form, the following remarks are inserted merely for reference, and not as a complete description for the instruction of the learner, who must acquire this knowledge with that of the rest of his duty. 404. The time in the ship's log-book is reckoned from mid- night, as civil or common time ; the first hour is, therefore, 1 o'clock in the morning, and the hours are carried on to 12, or noon, and then to 12, or midnight. The log-board, however, is copied into the log-book each day at noon.* 405. At noon, if the ship is in sight of land, a point or object of known latitude or longitude is set, and its distance estimated. This method of taking a Departure, which, from its convenience, is in general use (No. 349), is sufficiently accurate when the ship ia very near the land ; but when the land is distant, or enveloped in haze, and when, in consequence, the estimation of distance is liable to great uncertainty, some other method should, if prac- ticable, be adopted in preference, or at least employed as a check. If there is no particular object in sight, the extremes of the land iire set; and thus, in case of a fog coming on, the ship is secured, by keeping outside of the bearings of these extremes, from ap- jjroaching the Jand.f * The log-board, on which were painted the necessary divisions, and th;- record mado in chalk, has long passed away. A log-slate or deck log-hook is kept instead. T Since, when the ship is in sight of laud, hur place is determined with reference to the land alone, it is customary, during lliis time, to discontinue heaving the log, and thcreforo to omit the inscnion of the courses aiul distai.ecs on the log-board. It is .someiimes, however, proper to keep up the aceouut wlien in witli tlie land, as it alFords the means uf diseo\ering a permanent curriut, or the direction, streugth, and time of ehauge of Uid tidc-currenu THE ship's journal. 155 If tlio =(ii]i is out of sinlit of laml, tlie Course and Distance made 1(00(1 in tiie last 24 liours, the Latitude and Longitude by J3fud Hockoninfi:, as also by Observations if tlicy are obtained, are in- serted, together with the Bearing;, and Distance of the port or o/ the land worked for. 406. It often hap])ens, from change of long., that the day of 24'' has expired before the sun has attained the meridian. In this case, the hours having been truly measured, and the hourly distances rightly assigned, the reckoning is truly registered up to the run- ning out of the hist glass, and an increased distance must therefore ho marked against the last hour or half-hour. In like manner the day may really have expired by observation before the 24 hours are completed. In this case a diminished dis- tance must be marked at the last hour or half-hour. 407. The Leeway should always be marked on the log-board, since it is impossible for any one to know what leeway the ship may he making in bad weather when he is not on deck. 408. At the end of every watch, at tlie close and dawn of day, and at the coming on of a fog, the land is set; so that, in case of losing sight of it, a Departure may always be secured at the latest period. 409. The Weather is described at the end of each watch, or oftener, as occasion may suggest. In order to mark the strength of the wind, and the description of the weather, with more distinctness than the terms in general use among seamen are capable of express- ing. Sir F. Beaufort has proposed the following system of numbers and letters, which has been adopted by order of the Lords Com- missioners of the Admiralty, dated Dec. 28, 1838, in Her Majesty's Khips:— FiGURKS to denote the Force ok the Wind. — Calm. 1 — Light Air Or, just sufficient to give steerage way. 2— Liglit Breeze T Or, that in whicli a well-conili- I" i to 2 knots. - r. .1., li „ I tioncd man-of-war, witli all ) . , ^ 5 — l.cntlc Breeze ... ,• ... , , ',, , , , i to 4 knots. ' I sail set, and clean full, woultl ' ^ 4 — Moderate Breeze J go in smooth water from Ls to 6 knots. 5 .— Fresh Breeze ^ nioyals, Sic. 6 — Strong Breeze .... I Single- reefed toiisails ami tii|i- )0r, that to^ which she could) gallant sails. just carry HI chase, full and; „ 1, e ■, ^ -i ■■, . ', ' 1 Double-reefed topsads, Jill, I'^c Triple-reefed topsails, (<cc. (close-reefcd topsails and ccnirscs. re. — Whole Gale Or, that with which she could scarcely bear close-reefed main-top. sail and reefed foresail II — ."Jlnrm Or, that which would reduce her to storm-stavsaiU, IJ — Hurricane Or, that which no canvas codd withstand. 156 NAVIGATION. Letters to denote the State or the Weather - Rain continued i b — Blue sky; whetlier witli clear or hazy c — Cloudy ; but detached opening clouds. ri — Drizzling rain, f — Foggy — f, Thick fog. g — Gloomy dark weather, h — Hail. 1 — Lightning, ni — Misty hazy atmosphere, o — Overcast ; the whole sky being covered with an impervious cloud, p — Passing temporary showers. By the combination of these letters, all the ordinary phenomena of the weather may he recorded with facility and brevity. Examples: — b c m, Blue sky, with detached opening ilnuds, and a misty atmosphere, g v. Gloomy dark weather, but distant objects remarkably I'isible. q p d 1 t, Very hard squalls with passing showers of drizzle, and accompanied by lightning with very heavy thunder. t— Thunder. I — Ugly threatening appearance of the r — Visibility of distant objects, whelhei the sky be cloudy or not. f— Wet dew. . — Under any lettrr indicates an extraor- dinary degree. i-ell itlioul 410. When a lieavy sea is riimiing, or when corresponding wind, the circumstance is noted. A swell is named after the point of the com pass /?o»i which the waves proceed, like the wind tliat produces them. To denote, how- ever, a south-westerly swell (for example) as " a swell from the S.W " removes all ambiguity. 411. The variation of the compass, when observed, is inserted in the remarks ; as also the results of occasional observations, as the latitute by double altitude, by the moon, planets, or stars, the longi- tude by lunar, &e., the exact time of observation being specified. 412. In general, besides the details proper to the particular service on which a vessel may be employed, all matters relating to her place are inserted in the log, not only for the safety or conve- nience of the present voyage, but as matter of intelligence or of evidence in the case of future inquiry. Hence the circumstance of seeing or speaking a vessel is always noticed. No form of log has been universally adopted in merchant-ships, but several neat forms are in common use. The precise form is not material, as long as the ship's proceedings are exactly and conveniently recorded. A separate journal, called in the Eoyal Navy the engine-room register, is generally kept in steam-ships. In this is recorded the revolutions of the engines, the pressure of steam, the consumption of fuel and other materials, the temperature of the engine-room, stoke-holes, coal-bunkers, &c. Generally, it is a record of all matters relating to the performance and state of the engines, and the employment of the engine-rooin stuff. THR SHIP'S JOURNAL. l')7 41 3. The following is the form in which the logs of her Majesty's Bhips are at present kept by order of theEi>ar(lof Admiralty, 187!>. H.M.S. dayc f , 18 1 . 1 t..ilial3 of Officer V^atch 1 I s 1 standard Comiiass Courses .... 1 Veather III Height Remarks Dircc- Force Bar. =. 1 2 3 4 5 6 8 9 10 11 12 course B stance Latitude Longitude Variation allowed Water Remiiinf True Bearing and No. on Sick-l->t Curren mad goo mile the wale DR. Obs. DR. Chro. Daily Expend" ^ Distilled 8 miles tcnliiy 1 ,.«. 2 3 4 6 6 7 8 9 10 11 12 1 Coal expended Signnlsj diirnii; ( 24 hoiiri For enpnM For..l,ip r„r.l..^tillin» 15J n. The Day's Work. 417. This is the process of finding the place of the ship, with reference either to her place at yesterday's noon, or to a departure taijen since, and comprises, 1 st, The Course and Distance made good ; 2d, The Lat. and Long, in ; 3d, The Bearing and Distance of some port, which is either to be steered for directly, or is an intermediate point of land, with refer- ence to which the course is to be shaped, so as to make it or to avoid it. 418. To work a day's work. (1.) Take the courses, with the distance run on each, from the log-board. When a departure has been taken, consider it is a course anl distance in the opposite direction. Correct each course for deviation of the compass, 229, or p. 159. If the variation has changed since the departure was taken, cor- rect each course separately. No. 221; if not, defer this correction. Every course affected by leeway must be corrected accordingly. The quantity, if not marked on the board, must be estimated from the circumstances. When the ship is on the starboard tack, allow the leeway to the left ; when on the p07-t tack, allow it to the right, the observer being supposed in the centre of the compass. When the ship is hove-to, take the middle point between that to which she comes up and that to which she falls off, for the compass course, and correct this for leeway. (2.) Having corrected the Courses thus far, take out to each the D. Lat. and Dep. from the Traverse Table, and find the Course and Distance made good by Traverse Sailing, No. 287, or by Traverse Tables (Table I.) If the variation has not been allowed for, apply it to the result- ing course, No. 221. (3.) Apply the D. Lat. to Lat. left : the result is Lat. in, No. 190. With the Lat. left and Lat. in, and the Course, find the D. Long. by Case I. of Mid. Lat. or Mercator's Sailing (No. 315 or 323), or by Traverse Table. If the Course is due E. or W., then proceed by Case I. of Parallel Sailing (No. 304) or by Traverse Table. Having the Long, left and Diff. Long., find the Long, in. No. 1 9!). (4.) Having now the Lat. and Long, of the ship, and those of the port to be worked for, find its Bearing and Distance ; if in the Lat. of the ship, by Case II. of Parallel Sailing, No. 307 ; otherwise by Case IT. of Mid. Lat., or Mercator's Sailing, No. 318 or 326; or by Traverse Table. To this Bearing apply the Variation and Devia- tion of the Comx:)ass, and so obtain from the True Course, the Course to he steered. To find the Course on a Great circle, eee No. 337 or 338. It is mere waste of time to work the Course nearer than to tlie whole degree ; for even if the compass could be depended upon to 1^, the ship cannot generally be steered within that quantity. THE SHIPS JOURNAL. 159 Courses. U.U N. s. E. W. S.E. by S. 3 25 17 S.S.E. i E. 23 203 loS S.S.E. 49 45 3 18S S. by E. i E. 24 230 70 S. by E. 6 59 12 S.W. by S. S 67 44 S.W. 7 49 4-9 S.W. by W. 7 39 5-8 W. by N. S 10 49 S.iE. 6 60 06 uS-5 40-, 200 200 ■■75 20-. Kx. 1. The sliip while hove-to for the first two hours, with light nirth-eastorlj winds, came up to E., and fell off S.S.E. ; taking S.E. by E. as the niidillo cour.se, ;Ubiw- :ng 2 Jits, leeway, and 3 miles distance, gives S.E. by S. 3 miles, .iftor which the courses and dists. follow as below. Lat left 29° 26' N., long, left 127° 42' E. : Tar. 3° E.: find the Lat. and Long, in ; also set of current in the 24 hours. Position by observation being Lat. 27'^ 55' N., Lung. 128° 43' E. D. Lat, 118 i^sS'S. Lat. left D.R. 29 26 N. Lat. in, D.R. 27 28 N. Lat. left 29° and Lat. in 27° j^ive Mid Lat. 28°. Then 28° and D. Lit. 14-5 give Dist. 16' E. Long, left 127 42 E. Long in, D.R. 127 58 E. To determine approxiniale current soo Nos. 290 to 297, aud 1015. Position by Obs.Lat.27''S5'N.,Long. I2S''43' E. Position by D.R. Lat. 27 28 N., Long. 127 ;S K. 27 45 In Lat. 28° DifT. Long. 45 = L), p. 397- n-i r> T 1 _ . IT. ■ r. Then D. lat. 27 and Dep. -so 7 The D. Lat. 1 17-5 and Dep. 20-1 give Course „• „<-„,„„„ m -ao n n ^ d , rj c o i"; T. -1 e ves CounsE N. Co J^., Di:>t. 48 in.. by Compass S. 10^ E. L>ist. 1 19 mdes. , <•,. ■ u ■',',• .0/ \ . .1 ■ I. • ^ set of LuuHENT in 24 hours. Applying 3°(var.) to the right gives Course -n i 1 i ,v, I- v. ■ • .1 S. 7° E. true. Then 7° and Dist, 119 give ^ Prol.able; the ship being in the l). Lat. I i8-i, and Dep. 145. K;'"'° ^iwo, or Japan Stream. In the foregoing example, the deviation of the compass has not been mentioned. From what has been said in Chapter II. it must be evident that the bearing taken for departure and the courses steered must be corrected for deviation, where there is any. As the deviaiion changes when the direction of head is changed, it is obvious that ciih Course must be corrected separately. To correct the Compass for Variation or Devial ion. Course by Compas given. I True Counts yiven. If Var. or Dev. East, allow to riglit. If Var. or Der. E,ist, allow to left. If Var. or Dev. West, allow to left. If Var or Dev. West, allow to right. Will give true course. I Will give magnetic course. To Correct the Compass Courses. En-sterly Variation or Deviation is + to all points between N. and E S. and W. Westerly Variation or Deviation is — from all points botwi'oii N. and E....S. and W Easterly Variation or Deviation is — from all points between N. and W....S. and E. Westerly Variation or Deviation is + to all points between N. and W S. and E. To Convert a True Course or a Correct Magnetic Course into a Compass Course. En-sterly Variation or Deviation is — from all points between N. and E....S. .md W. Westerly Variation or Deviation is + to all points between N. and E S. and W. Easterly Variation or Deviation is + to all points hetween N.and W S. aud E. Westerly Variatiou or Deviation is — from all points butwoen N.and VV....S. and E. In the following examples the Deviations from table of No. 227 have been applied to the Coinjiass Courses, to obtain the Correct Magnetic Courses. ICO THE SHIPS JOURNAL. Ex, 2, The Departure is taken from the Edcljstone, bearing N.N.E. 12 miles. Ship'i head S. by E. Toe ship ran S. by E. 14 (miles), S. by W. to, and S.W. by W. 8. Allow 25° westerly variation. Find the Bearing and Distance of U-^hant, and Course to be steerud.* The Departure gives a Course S.S.W. (No. 418 (I)). Correetin:; this and the other Courses from the Deviation Table, No. 227, S.S.W. becomes S. 18^ \V. (No. 228), S by E. become.s S. 16° W., S. by W. becomes S. 6° W.; and S.W. by W. becomes S. 50° W. Eddystone Lat. 50° 11' N. l>- Lit. 40_^-__ Lat. i.v, D.R. 49 31 N. Lat. left 50° and Lat. in ^gl" give Mid Lat. 50°. Then 50° and 10 6 as D. Lit. give Disf. 16', the D. Long. Eddj'stone Long. 4° 16' W. I). Long. 16 E. Long, i.v, D.R. I o^W. Lat. in 49° 31' Long. 4° o' llshant 48 29 54 I 2 = 62' I 4 = 64' Mid. Lat. 49°. Course 49° and Dist. 64 give D. Lat. 42 ; this, as Dep. and I). Lar. 62, give Bearimg S. 34° W., DisT. 75 m. Compass Courses. Dists Correct K. S. E. w. S.S.W. 12 S. 18° w 114 ,37 S. by E. 14 S. 16 E '35 39 S. by W. 10 S. 6 W. 9-9 10 .S.W. by W. s S. 50 W. SI 39-9 39 61 108 3-9 6-9 D. Lat. 39' 9 and Dep 6 9 give Co. S. 10° W., DisT. 41. Applying 25° to the left gives CooHSF S. 15° E. true. Then Ccnrse 15° and Dist. 4t give D. Lat. 39 6 and Dep. 106. Then Course S. 34" W. + Var. 25° W. gives S. 59° W. + Deviation 7° W. give S. 66° W., Course to be steered for Ushant. Ex. A ship from lat. 0° 5' N., and h ng. 0° 17' W., sails S.W. by S 7 miles, S. by E. 22, S.S.W. i W. 8, and N.E. by E. 20. V.ir. 19° W. Position by Obs. Lau 0° 15' S., Long. o°2oW._ Find Compass Course to be steered,* and tlie Dist. to"C. Palmas ; sperieneed in the 24 hours. Compass Courses. Dists. Correct Magnetic Courses. N. s. K w. S.W. by S. 7 S.29°W. 61 34 S. by E. 22 S. 16 E. 211 61 S.S.W. AAV. 8 S. 23 w. 7 4 3« N.E. by E. 20 N.70 E. 68 188 68 34-6 24-9 6-5 1 68 65 278 184 D. Lat. 278 and Dep. 184 give Co. S. 33° E., Di.sT. 33 miles. Applying 19° var. AV. to the lelt, gives Course S. 52° E. true. Then Course 52° and Dist. 33 give D. Lat. 20-3 and Dep. 26. To determine approximate Current, see Nob. »90 to 297, and 101.5. Lat. Obsd. 0° 1 5' S. Long. 0° 20' W. Lat. D.B. o 15 S. Long, o 9 E. Approximate Cureent West 29 ra. Lat. from 0° 5' N. h. Lat. o 20 S. LiT. IN, D.R. o IS Si Near the equator Dep. is D, Long., No. 311 ; hence, Long, from 0° 17' W. D. Long. 26 E. Long, in, D.R. o g E. By Obs. Lat. 0° 15' S. Long. 0° 20' W. C. Pal. 4 22 N. 7 44 W 4 37 = 277' 7 24 = 444- D. Lat. 277 and Dep. 444 give Course N. 58° W., and Dist. 523 miles; Course, N. 58° W. Thou -Var. 19° W. = N. 39° W. -Dev. 8 W. = N. 31 W., Compass Course to be steered. N.B. — On this course allow for cros.sing the Equatorial and Guinea Currents. * In shaping the Course, consider the direction and force of the tide may be found, tetnten the position of the ship and the port steered for. NAUTICAL ASTRONOMY. CHAPTER 1. Definitions. 419. This branch of the subject, as already tlcfinetl utkIcf the lioad Navijration, No. 179, relates to finding the place of the spec- tator on the surface of the earth by observation of the heavenly bodies. 4'20. To the spectator at the surface of the earth the heavens dppear to form a vault, or the upper half of a hollow sphere, of which he is the centre; the earth itself, or the ground or sea on which he stands, occupying the lower half. Any two points on the apparent concave or celestial surface, as two stars, for example, may be sup])osed to be connected by an arc of a circle drawn on that surface : and tiius the apparent celestial sphere may be conceived tn be marked with circles like the terrestrial globe. 421. The spectator stands with his feet towards the centre of the globe; that is, a plumb-line, which is vertical, passes through the spectator and this centre;* and thus the spectator always conceives himself on tlie summit of the globe.i' Suppose him now to descend the above line to the centre, and then suppose the upper half of the earth or globe to be cut off horizontally, that is, parallel to the horizon, or perpendicular to the plumb-line. The surface of the; lower half-globe, or hemisphere, so e.xposed, being produced on all sides to meet the concave celestial surface, is called the Rational • The earth is here supposed to he a globe ; the plumb-line does not exactly pass throujh the centre of the spheroid, but the difference is not worth notice here. t This is the principle of rectifying the globe, or placing the globe to shew the relatife poflition of the spectator and the heavens. To rectify the globe, as, for ex., for Greenwich, in 51° N. Lat. Place the globe on a level surface, so that the broad rim, or horizon, shall be horizontal. Take hold of the brass meridian, and turn the globe round in its stand (upwards or downwards) until the N. pole is 51'^ above the rim. Direct the N. point of the rim (now under the pole) to the true north. Turn the globe round its axis till Greenwich passes under the meridian j Greenwich will now be the upper- most point. The axis of the globe now makes the same angle with the wooden horizon thai the axis of the beaveus (or line joining the centre and the poles) makes iviili the horizon of the i\tee- tntur. M [G2 NAUTICAL ASTRONOMY. HonizoN. Every point of the eartli's surface lias thus a difierei.t rational horizon, but all these horizons have the same centre. 422. It becomes, in general, necessary, for considerations whicl'. will appear hereafter, to reduce celestial observations taken at the surface of the earth to what tney would have been if taken at tiie centre; in the following figures, therefore, the observer is supposed to be at the centre of the earth. The dimensions of tKe earth are so small in comparison with the vast distances of the stars, that the above change of place of the spectator from the surface to the centre, or to any other point, would produce no change whatever in the apparent places or directions of the stars ; and, accordingly, the magnitude of the earth, in drawing figures for general purposes, is neglected, the earth itself being considered as a mere point in the centre of the great sphere which circumscribes the stars. In the case of nearer bodies, as the sun and some others, and especially the moon, which, when viewed with delicate instruments, appear in ditlerent directions when seen from different points of the surface oi the earth, this apparent change of place is allowed lor by a special calculation. (See Parallax, No. 435) 423. The Zenith is the point vertically over the spectator, and distant 90° from the rational horizon at every point. The point opposite the zenith, or under the spectator's feet, on the other side of the centre, is called the Nadik. In fig. 1, NWS E represents the Rational Horizon; N S, the Meridian of the observer; N, S, E, \V, the North, South, East, and West points ; Z, the Zenith, which is seen directly over, or in one with the centre. This figure is drawn on the plane of the rational horizon, and shews the several circles as they would appear to an eye looking down vertically from a point at a great distance above the zenith. Fig. 2 is drawn on the plane of the meridian, and shews the several circles of the upper or visible half of the sphere, as they would appear to the eye situated at a great distanc3 due east of the sphere. In this figure the circle N WS E, or the horizon, appears as a straight lin» N S being seen edgeways ; while the meridian. DFTINfTIOXE. ] (J3 whicli ill ri;,'. 1 is tlie straiglit line N S, appears Iiere as tlie seiuici-cle N P Z S. Tlie E and W points are seen in one with the centre. Of these two figiirrs, that one woiiicl naturally be preferred which would hest illustrate a proposed case. Fig. I may generally be employed to exhibit the hour-angle and azimuth; and fig. 2 the altitude, when the celtstial body is near the horizon.* 424. P, the Pole of the heavens, is the point which remains fixed, whilst the rest of the celestial surface seen above the horizon appears to revolve. The pole P is here represented as the North pole ; the other extremity of the axis round which tiie sphere appears to revolve is the South pole, and takes the place of P when tiie figure is drawn for S. Lat. This pole is called the elevated pole. 425. The circle E M W, 90° from the pole, is the C'elestiai. Equator. The plane of the earth's e([uator, EMW, fig. p. .'j.5. No. 180, being extended to the heavens, marks on the sphere the celestial equator. 426. A Celestial Meridia.v is a semicircle passing through the pole of the heavens; PZS is the celestial meridian of the spectator. The ))lane of the terrestrial meridian extended to the heavens marks on the sphere the celestial meridian. 427. Circles of Altitude are circles passing through the lenith, and vertical at the place of the spectator. Thus Z A H is the circle of altitude jiassing through a star A. Such, also, are ZMS, ZPN. 428. The Prime Vertical is the vertical circle E Z W passing through the E. and W. points. In fig. 2, EZW does not appear, being in one with C Z, a radius joining the centre and zenith. When the observer is on the equator, the celestial equator and prime vertical coincide. 429. Altitude is measured on a circle of altitude from the horizon ; thus A H is the altitude of A. The arc A H is the measure of the angle A C H, which would be formed at the centre by two straight lines, C H and C A. The alt. of a body M on the meridian is M S, which is the measure of the angle M CS. 430. Parallels of Altitude are circles parallel to the horizon. 431. Zenith Distance is the arc included between the zenith and the celestial body, or the angular distance of a body from the zenith of which that arc is the measure. The zenith distance is, therefore, the complement of the altitude to 90°, as Z A. 432. The altitude of a celestial body, as seen from the surface of the earth, is called the apparent altitude ; as seen from the centic, die true altitude. A ray of light, proceeding from the body, when not in the zenith, to the eye, in traversing the earth's atmosphere, which is heavier, or denser, as it is nearer the surface, is bent more and more as i< • In like manner thf. figure may be drawn in the plane nf t!v mritor (n» in No». Ot, nt), in that of the prune verlicd, or any other circle. IGl KAUTICAT, ASTRONOJlTf, approaelies the earth, towards the perpendicular direction ; and as the spectator sees any object, not always in its true direction, but in that direction in which the light from it finally enters his eye, a celestial body appears higher than its true place. Thus, the ray S A, wiiich proceeds from a star, is more and more bent towards the vertical line A Z as it approaches the surface, whereby the spectator sees the star in the direc- tion A S', and therefore higher than its true position. The ray A Z, which traverses th( atmosphere perpendicularly, undergoes „^'. uo refraction. Thus to the eye supposed -^^^..J^ ^^^^^"^^"^a^ at the centre all rays would proceed — -^-^ without any deviation ; because lines drawn towards the centre of the sphere are perj>endicular to its circumference, parallel to whicM the atmosphere is disposed. 433. This alteration in the apparent place of a celestial body, caused by the atmosphere, is called the Astronomical Refraction. The astronomical refraction is at the zenith, and about 34' at the horizon ; hence a celestial body, when really on the horizon, appears elevated 34' above it, and is seen on the horizon when really 34 below it. From the same cause all the celestial bodies rise earlier and set later than they would were there no atmosphere. The refi'action varies with the density or weight of the air, being greater when the barometer is high, or the air cold, and less when the barometer is low, or the air warm. The m^an refraction, or that in the average state of the atmosphere, is given in Table 31, and corrections for different states of the air in Tables 32 and 33. Since refraction causes the object to appear too high, it is to be sub- tracted from the apparent altitude in reducing it to the true altitude. 434. Twilight is the effect of the illumination of the upper regions of the atmosphere by the sun, before he has risen or after he has set, at the place of the spectator. Twilight continues, gene- rally, while.the sun is less than 18° below the horizon. 435. Parallax in Altitude is the angular depression of a celestial body, in consequence of its being seen from the surfaco instead of the centre of the earth, thus: DEFINITIONS. IC^ llie body S, wliicli is vertical to the spectator (wlio always stands .vitli his feet towards tlie centre) at B, in the line C S, appears at T, being seen in the direction CST; while to a spectator at A the same body appears below T at U, or in the direction A S U ; the angle A S C, or T S U, which is equal to A S 0. No. 1 16, is tUeparulltui in altitude. (Tables 34 and 45.) The spectator at B sees S in the same line as if he were al ilie centre; that is, a body in the /.enith has no parallax. To a s])ectator at D, to whom S appears in the horizon, the depression, or parallax, is greater than at any other point. The parallax at the horizon is called the Horizontal Parallax. Since parallax makes the object apj)ear too low, it is to be added to tlie apparent altitude, in reducing it to the true altitude. 436. It is evident, by the fig. No. 435, that the farther off a celestial body is, the less parallax it will have ; and the nearer, the more. The sun has about 9" lior. par. : the moon has about 1*. I'arallax is matter of actual observation, and determines definitively the distances of the sun, moon, and planets. 437. The parallax will obviously be less if the eartli's radius is less. Now, the earth being shaped like an orange, the radius, or line from the centre to the surface, in any latitude, is less than at the equator ; hence the moon's hor. par. in the Nautical Almanac, which is the eqimtoreal hor. par., is too great for any latitude. The reduction is given in Table 41. 438. Since the apparent altitude is too great on account of refraction, and too small on account of parallax, the diff. between these quantities is the diff. between the true and apparent altitudes. This difference, or the combined effect of parallax and refraction, is called the Correction of Altitude. The moon's Corr. of Alt. is given in Table 39; that of a star is merely its refraction. 439. The Semi-diameter of a celestial body is half the angle subtended by the diameter of the visible disc. Tims to a spectator at S the semi-diameter of the body is half the angle subtended by the diameter D F, or contained between the lines S D, S F, supposed to be drawn from S to D and F; the half of this angle is D S C or C S F. and is called the seuu-diameter. It is evident that the semi-diameter will be greater as the body is nearer, and smaller as it is farther off. Thus the variations in the semi-diameter of the sun ])rove that the distance between the Bun and the earth varies at different times of the year. (Table 34.) 440. When the body S is in the zenith, it is nearer to the 8i)ec- tator by half the earth's diameter, C B, than when it is on the horizon ; hence it ai)pears larger when in the zenith. This increase of a})parent dimensions due to increase of altitude is sensible in the case of the moon only, and is called her Augmentation.* This Is given in Table 42. • The »pparent increase of the magnitiidcs of the sun and mot b B m".re optica; illusion, whatever explanation may be given of it ; IGR NAimCAI. ASTRONOMT. 441. The Declination of a celestial body is tiie portion of the nieridiaii between the equator and the body ; it is reckoned from the equator, and is either north or south. Thus, A B, tiff. 2, p. 162, i* the Declin. of A, and is north. Since the declination is measured on the celestial meridians, these Are called also declination circles. 442. Parallels of Declination are circles parallel to the equator, as the dotted line through A, in both figures, p. 162. Thus declination is reckoned from the celestial equator as latitude en the surface of the earth is reckoned from the terrestrial equator; and as both these circles are in one and the same plane, declination and terrestrial latitude corres])ond : that is, a star in 28'*N.Decl. jtasses every day vertically over all places in 28° N. Lat. 443. Polar Distance is the arc of the celestial meridian between a celestial body and the pole, or the angular distance of a body from the pole. When the Lat. and Ded. are of the same name, the pol. dist. is the compl. of the Decl. to 9U'', because the distance I'rom the pole to the equator is 90°; when the lat. and decl. are of different names, the pol. dist. is the sum of the decl. and 90°. Thus the pol. dist. of A is PA; tliat of A' in S. decl., fig. 2, is P A', which is the sum of 90° and A'B. 444. The Azimuth of a celestial body is the angle at the zenith contained between the meridian of the place of the spectator and the eii'cle of altitude passing through the body. It is reckoned to begin from that part of the meridian which is on the polar side of the zenith, that is, from the N. in north latitude; thus, the angle PZA is the azimuth of A. The angle MZA is the supplement of the azimuth to 180*. This is often used for convenience; thus, instead of N. 132" E., we say S. 48» E. 44,5. The angle N Z A or P Z A is the same thing as an angle N C II on the horizontal plane, contained between tiie north and south line C N, and a line from the eye at C to the foot of the circle of altitude H,* which is the "point of the compass" on which A is seen. Now the angle N C H is measured by the are N H ; the azimuth, accordingly, is measured by the arc of the horizon between the meridian of the place and the circle of altitude of the body. 'Jhe ship's course is the azimuth of the ship's head ; so, also, the hearing of an object is its aziinutii ; and difference of bearing is difference of azimuth. When a body is on the prime vertical, its azimuth is 90". Since refraction and parallax take place vertically, they do not affect tiie azimuth of a body. 446. The Amplitudi;; is the arc of the horizon between a celestial body at rising or setting and the E. or W. point, and is the coin- irhich the angles subtended by the discs are measured discover no change of magnitude. The constellations, as the Great Bear, Orion, &c., appear in like manner, when near the horizon, to occupy a vast space in the heavens, but when near the zenith much less. * This cannot be distinctly represented to the eye by figs 1 and 2, because in fig. 1 lb* poiuts Z and C coincide, and in fig. 2 the horizon N W S E appears as a straight line UEFINITIONS, 1G7 pleiiUMit ot tlie aziinntli ; thus E H is tlie !iiii|>litiKle of ii body risiiij; Rt II. Aiuplitiide is reckoned from the E. or \\ . ; thus, if E H is •27* the auiidiuide of H is E. 27° S. (I.) The great refraction at tlie liurizon affects sensibly the appa- rent aniplitiide. Thus, suppose the spectator in north hit. facing the I'ast, EQ part of tlie equator, EZ part of tlie prime vertical, A' a star having north deel. then E A' is the apparent amplitude at the instant of rising; but the star is known to be raised, that is, brought into view, in this case, by refraction, and therefore has not yet, in its revolution, arrived at the horizon ; A' is consequently to the left of the place A, where it would rise were there no atmosphere. Hence the arc A'A is applied to the right ' of the compass-bearing on which A' is observed, in order to correct the apparent place of tlie star for the eti'ect of refraction. This quan- tity is given in Table 59 A. In facing the west the line EQ (which would become W Q) would lie on the other side of the prime vertical, and the star would be seen to set to the riylit of its true j)lace. In south lat. the figure drawn above answers to setting, putting ^y. for E. (2.) As the elevation of the observer depresses the sea-horizon wliile it does not affect the place of the star, it produces a further effect of the same kind as that of refraction. In the case of the moon, as her parallax exceeds the refrac» tion, the opposite effect is produced ; that is, when she appears to rise, she has already, to an eye at the centre, passed the rational horizon : thus A would be the apparent place of the moon at rising, to the ritjhf. of the true place A'. 417. The latitude, or distance of the observer from the equator, is measured, on the celestial sphere, by the distance of his zenith from the celestial equator ; or Z M is the measure of the latitude, figs. p. 162. Suppose now U, a star of N. dccl., on the meridian at D., then M D is its decl. and Z D its zenith distance ; here Z M, the Lat., is the sum of the decl. and zen. dist. If D' be a star of S. decl., Z M is the diff. of Z D' and M D'. If a star d be between Z and P, the lat. Z M is the difference of M (/ and Z d. 448. Wl-.en the object is to tlie south of the observer, that is, when his zenith is to the north of the body, the zen. dist. is com- monly called N.; when his zenith i^j to the south of the body, the zen. dist. is called S. In fig. 2, ZD and Z D' are therefore called North, Zd is called South. It appears, hence, that when the Decl. and Zen. Dist. are of (he inme name, their smm is the latitude; when o( different names, their difference is the latitude. But when the star is below tlie pole, as at d', the Lat.ZM is 1C8 JAUTICAL ASTRONOMY. the Dlff. jf Md' and Zd', and M d' is the sum of M P and Pa* or of flu", and the couipl. of the dfcl. 449. M Z bein<r the lat , P Z is tlie Colat., since P M is 90". Also Z N beinj,' 90®, P N is the conipl. of P Z, and therefore equal to M Z ; or the elevation of the pole is equal to the lat. of the place. 450. The altitude of the uppermost point of the equator on the meridian, or MS, is equal to the cohititude, because Z S is 90°. By noting this, and also that the equator passes through the E. and W. points, it is easy, in looking towards the heavens, to figure in llie inind, roughly, the position of this circle. Tiiis is often useful. 451. In high latitudes, P in tlie figure falls near Z ; in low lati- tudes, P falls near N. On the equator, Z and M coincide, the celestial equator there passing over the spectator's head. In S. Lat. the letters N and S in the figures are changed; also tlie direction of the celestial motions (which we in N. lat. consider from left to right) is there reversed, because in S. lat., in looking towards the equator, the E. is on the right hand. 452. By the help of the preceding considerations (No. 447 and following) it is easy to construct a figure, in any case, to exhibit at o?ice the manner in which the latitude is obtained from the meri- dian altitude and the declination. Fig. 1. Fiff. 2. Fi!/. 3. Ex. 1. The Mer. Alt. of the sun, observed to the southwarti, is 58° ; liis Decl. 14° N Fig. 1. Draw a quadrant Z C S by means of the chord of 60° (No. 107 ). Lav off, by the scale of chords, the Alt. S ©, 58°, or the zen. dist. Z 0, 32°. Lay off the Decl. 14° to the »oulhward of the sun, as © M, since he is to the northward of the equator; then !\I is on the equator, and Z M is the Lat. nnrtli, and measures 46°. Ex. 2. The Mer. Alt. of the sun, south of the observer, is 19°; his Decl. 18° S. Fig. 2. Lay off S © , 29", and ® M, 1 8° to the N. of the sun ; then M is the place of tha equator, and Z M, the Lat. north, measures 43°. Ex. 3. The Mer. Alt. of the sun, north of the ooserver, is 38°; his Decl. 14=^ N. Fig. 3. Lay off N©, the Mer. Alt. 38°, and ®M the Decl. 14° to the S. of 0; thsu £ M is the Lat. south, and measures 38°. These figures, which are varieties of fig. 2, p. 162, are of the simplest kind. The point Z being marked on the quadrant, the place of the sun at Q, north or south of the observer, is given by tlie obser nation ; his declination gives M the place where the equa- tor cuts the meridian ; whence it is at once seen whether Z is north or south cf M, that is, whether the Lat. is N. or S.* * After a little practice the observer will perceive, at the time of observation, h deduce the latitude from the mer. alt. and decl. independently of the distictions of." sJxjve (No. 448), which are adopted for the purpose of forming a general rule. DEFINITIONS. 1 G9 ■153. Tlic passage of a celestial body over any particular point or circle is called TRANsrr ; as the transit of the meridian, or the prime vertical, of a planet over the sun's disc, &c. 454. Culmination is another term for transit of the meridian. The transit of the meridian below the pole, whether above or belovr the spectator's horizon, is called the lower culmination ; the other transit is called the upper culmination. 455. OccHLTATioN is the disappearance or hiding of a celestial body by tiie intervention of another. Thus the stars in the moon's path are occulted by her, and the satellites of a planet by the body of the planet. 456. Eclipse is the disappearance of a celestial body in the shadow of another. In an eclipse of the moon, she disappears wholly, or partly, in the shadow of the earth, the earth being then in a line between the sun and moon. In an eclipse of the sun, the moon, being then in a line between the sun and the earth, conceals from us, for a time, tne whole or part of the sun. 457. Celestial bodies are said to be in Conjunction when in a line together, as seen from the centre of the earth. Bodies having the same Right Ascension are said to be in Conjunction in Right Ascen- sion (No. 469). Two bodies are said to be in Opposition when in diametrically opposite points of the heavens. 458. It will be perceived, on attending to the circumstance, that stars which are visible in the west soon after sunset, disappear after some days in the solar light; and, in like manner, that stars which are faintly seen in the east, before sunrise, become more distinct from day to day. Hence the sun, besides revolving daily with the fixed stars* from east to west, has an apparent yearly motion amongst them in the contrary direction, or from west to east, com- pleting the circuit of the heavens in the course of a year. 459. The ])ath on which the sun appears to move, or the great circle which he seems to describe in the heavens, is called the Ecliptic. 460. The ecliptic is divided into twelve Signs, or portions of 30^ each, called the Sifins of the Zodiac, which term originally meant a space or belt of 8® wide on each side of the ecliptic, to which the planets-f- are confined. The signs, taken in the order in which the * Tlie stars are bodies which shine by the'.r own light, and astronomers conclude, frnm every analogy yet detected, that they are suns. They are called " fixed," because to tUs eye they appear always in the same relative positions with respect to each other. The distance of the stars is so great that the difference of angular position, as seen from opposite points of the earth's orbit, a distance of a hundred and ninety millions of miles, has been found, in the case of one star only, to amount to so large a quantity as 2", according to Mr. Henderson's determination of the parallax of a Centauri. At this star, therefore, the sun, which to us appears under an angle of above half a degree, would subtend an angle of or.ly two hundredths of a second. + The planets are bodies which, like the moon, shine by light received from the sun and rellpcted to us ; they revolve round the sun in the same direction as the earth, bnt in different periods of time. Mercury 5, the nearest to the sun, revolves in 88 days; Venns ?, the ncit, in 225 days. These, moriug ia orbits iuside ihat of the Earth, arc called inferior 170 NAUTICAL ASTRONOMY. 5!in moves througli tbeiii, that is, in tliC contrary direction to t).e apparent diurnal motion, are as follow : — <Y> Aries (the Ram). I £= Libra (the Balance). 8 Taurus (the Bull). m Scorpio (the Scorpion). u Gemini (the Twins). ] f Sagittarius (the Archer). ffi Cancer (the Crab). I yf Capricornus (the Goat). SI Leo (the Lion). ^r Aquarius (Water Bearer), njl Virgo (the Virgin). | x Pisces (the Fishes). 4HI. Besides this perpetual motion from west to east, the sun is always changing his declination, which varies between 23° 28' N. und 23° 28' S. He crosses the equator twice in the year, namely, about the 20th of March, in coming up to us in N. lat. from the southward, and again about the 23d of Sept. in going to the souta- ward. 462. When the sun crosses the equator, he rises and sets at six o'clock in all parts of the world;* at these times, therefore, the days and nights are every where equal. 463. The two points in which the ecliptic, or sun's path, thus cuts the equator, are called the Vernal, or spring. Equinox, and the Autumnal. Equinox. 464. The sun attains his greatest N. decl. about June 21st, and the greatest S. decl. about Dec. 22d. The points at which the sun seems at these times to be stationary in declination before he dimi- nishes it, and at which the ecliptic and equator are most widely separated, are called the Summer and Winter Solstices. 465. As the light and heat received from the sun at any place vary with his altitude, and the time during which he remains above the horizon, and as both of these depend on the declination, the succession of seasons depends on the changes oftlie declination of the sun. The common or civil year, as most convenient for the affairs of life, includes the succession of the seasons. It is, therefore, the interval in which the sun leaves any parallel of declination and returns to it again, and is called a tropical year. Its length, that is, the average length of a number of such years, is 365'' 5''48"61'-6, of common or mean tiuie.f planets. Mare <J revolves in nearly 2 years; Jupiter 2f , in nearly 12 years; Saturn h, in 29 years; Herschel Ijl, in 82 years ; and Neptune ^, in 165 years. These list are called superior planets, Be-ides these there are numerous small planets [287 known in 1890] whose orbits lie between tho^e of Mars and Jupiter. Some of the pl.inets have Battllites, or moons : Mars has two, Jupiter four, Saturn eight, Herschel six, and Neptune one. * The observed times differ a little from 6'' on account of re''raction. No. 44H. •f If the tropical year eontainrd exactly 365 days, the arrangement of the cali-ndar would be perfectly simple ; but the necessity of counting by entire days in the iiffairs iif life has introduced arbitrary expedients for checking the errors accumulated from time to time, from neglecting the excess over the last complete day. For example, suppose tlio year ends iit midn ght on Thursday, then new year's day begins at the same instant, that is, at 0'" on Friday morning, while the old year is really not yet out by nearly 6 hours. Next year 6 hours more of the new year will be anticipated, that is, new year's day will be reckoned 12 hours too soon ; so that at the end of 4 years the beginning of the new year is antici ate 1 by a whole day. By adding 1 day to the fourth year this error is re- moved, and the cojnnicncement of the calendar year is carried back to its true place nearly DEFINITIONS. 171 The peiiod of the conmienceinent of the year, whicli has been »(loi)tecl dirt'ereiitlj' at different times, is at present (as establisiied in this country by act of parliament) on January 1st, which is about 11 days after the winter solstice. 466. Since it is summer on that side of the equator on which the fuii is, and winter on that on which he is not, the seasons in south latitude are reversed. 467. In the continual apparent revolution of tlie lieavens round the earth, the circles of declination are perjietually describing angles round tlie poles, which are called, from the division of time into hours, Hour-Angles. 468. An hour-angle, or horary angle (sometimes called also Me- ridian Distance), is the angle at the pole contained between the meridian of the place and the celestial meridian passing through the body; thus, ZPA is the hour-angle of A (figs. p. I6i). An hour- angle. is measured by the arc of the equator contained between the meridian of the place and that of the body; thus MB, fig. 2, measures ZPA. The hour-angle is thus measured on the celestial equator in th« same way as longitude is measured on the terrestrial equator. 469. The Right Ascension of a celestial body is the arc of the equator included between the first point of Aries and the celestial meridian of the body: it is reckoned from west to east. Thus, if rr be the first point o{ Aries, fig. 1, p. 162, the arc t MB is the Rigiit Ascension of the body A. The 360° of the celestial equator are divided into 24" of R.A. Thus R.A. is reckoned on the celestial equator exactly as the longitude of places on the earth is reckoned on the terrestrial equator. But as the stars do not preserve that constant position with respect to the meridians which they do with respect to the equator, there is not that corres])ondence between R.A. and longitude which there is between declination and latitude. 470. The apparent revolution of the stars is perfectly regular, and is the only motion of the kind known. One revolution of the earth round its axis, or, which is the same thing, the return of the same fixed star to the meridian after com- pleting the circle, constitutes a sidereal day; this day consists of 23'' 56™ 4' of common or mean time, as measured by clocks and watches. It is divided into 24 hours, called sidereal hours, and these into sidereal minutes and seconds. Thus a sidereal day is about 10* But the excess above 365'' does not amount to 6* by ll" 8' nearly; hence at the end of the fourth year an error of the contrary kind is introduced of 44" 32', which amounts to nearly 3 dnys in 4 centuries. This error led to the reformation of the calemlar by I'upe Gregory XIII., in I.i82, when the vernal equinox, which at the Council of Niie, in 32.5, had laken place on the 21st March, fell on the 11th. Hence, leaving 10 days out of the calendar, v»biih was effiCted by calling the 4th of October, 1582, the 15ih, broug t inatiers right again. The error had amounted to II days when the change was adopted in this country in 1751. This error is prevented for a long period in future by the Act 24 Geo. II., which directs the Icap-yearg 1800, 1900, 2100, and so on, to be considered as common ytars, and 2UU0, liuo, 2800 as leap-years. 172 NAUTICAL ASTRONOMY. an hour si orter than a common or mean day ; and the sidereal liours, minutes, and seconds, in the same proportion. The sidereal day being thus, in round numbers, 4"" shorter tha.i the mean day, a star that passed the meridian last night at 9 p.m will pass this evening at S"" SG"", and so on, till after a few months il will pass at noon. (See Table 27.) 471. Sidereal Time begins (tliat is, a sidereal clock, regulated to sidereal time, shews 0'' 0'" 0") when the first point of Aries is on the meridian, and is counted through 24 hours, till the same point returns again ; the hour-angle of this point is accordingly sidereal time. The hour-angle of the first point of Aries is the right ascension of the meridian, No. 469, which is accordingly sidereal time. Dif- ference of R.A. may, in like manner, be considei'ed as a portion of sidereal time. 472. P is the pole, the circle N W M E the celestial equator, to which the mea- sures of all hour-angles are referred. Tiie bent arrow shews the direction of the apparent diurnal motion of the ce- lestial bodies, reckoned from east to west supposing the spectator to face the south. M N is the observer's meridian. A is any celestial body, as a star, which has passed the meridian at M, then APM is the hour-anyle of A, of which the arc AM is the measure. (1.) B is a star to tiie eastward of the meridian, which it has passed at N; its hour-angle, reckoned westwards, is measured by MWNB. We may, however, employ also BM, the measure of the hour-anglo reckoned eastwards. Thus, instead of 14*" 11™ W. we may call it 9'' 49"" E. As in dealing with hour-angles we refer di- rectly to the number of hours which they contain, and which are measured on the equator, it is unnecessary to form the hour-angle oi B by joining B and the pole. (2.) Let the first point or beginning oi Aries be at (y, having passed the meridian before the star A; then "r M is the riyiit ascen- sion of the meridian, that is, sidereal time. The R..\. of A is <r A ; that of B is <rMB, reckoned always from west to east, or opposite to the diurnal motion ; and <r N B is the supplement of the R.A. of B to 24 hours. (3.) The sidereal time tM is the sum of the arcs <r A and A M, that is, of the hour-angle and R.A. of the star A. Again, -rM is the difference between the arcs aM and wr, that is, between the hour-angle of the star a and the supplement of its R.A. In the case of the star B, tiie sid. time is the difference between its R.A. tMB, and its hour-angle M B. Hence it is easy, when the hour-angle of a star of known R.A. is given, at any instant of time, to construct the figure to shew the sidereal time, thus: — Having drawn a circle, with the meridian, l;vy DF.PIXITIONS. 173 otV, by a scale of chords, tlie star's iiour-angle ; tlie jio.sition of tlie star being now given, lay oft" it's 11. A., reckoning from the xtar in the same dirtction as the apparent dinrnal motion (for thus the 11. A. reckoned back again from tliis point <v will agree with the place ol the star). This gives the place of t, the hour-angle of which, reck- oned westward, is the sid. time required.* Ex. 1. The hour-angle of a slar is a"" iS"' W. ; its R.A. j"" 47"". Lay off 2'' zS'", or 37°, to the W. of M, and 3'' 47"', or 56° 45', further en towards tli« WO»t: tlien the sid. time measures 93° 45', or 6'' 15"'. Ejc. 2. The hour angle of the moon is t)^ 13"' W. ; her R.A. iS*" 34". Lay off 61^. or 90° (No. 107), and 3I' n"", or 48° 15', from M, westwards. Then lay off 3 times 6**, or 90°, and 34"*, or 8^ 30', further: tiie sid. time measures 56^45', or 3*^ 47™. Ex.3. Tlie hour-angle of a star is 14'' 1 1"' W., V 9'' 49"" E. ; its R.A. s""!!". The sid. time is 19'' 31™. All honr-angles, which are differences of R.A. of ihe meridian and a celestial body, may be considered as portions of sidereal time. The interval of time in which a body of variable R.A. describes an hour-angle depends on the rate at which its R.A. changes. 473. The earth's motion round its axis being perfectly uniform, becomes the real standard of uniform measures of time ; but as any Ftar passes the meridian nearly 4"" earlier every night, the beginning of the sidereal day has no connexion with that of the common or civil day, as determined by light and darkness. 474. The hour-angle of the sun, reckoning always westward from the meridian, is Apparent Time. Thus, when the sun's meridian has passed over 48* of the celestial equator to the westward of the meridian of the place, it is said to be 3'' 12™ apparent time. This is the time shewn by the sun-dial. 475. The interval between the sun's passing the meridian on one day and the next, or the apparent solar day, is not always of the same length, the difference being sometimes half a minute between one day and the next. A))paront time serves well enough in cases where this irregularity does not appear, or is of no importance; as for ex- ample at sea, where, from the continual change of longitude, the time must be obtained by observation : but where account of the time is to be kept by mechanism alone, it must necessarily be divided into portions of invariable length. The time for general use must, accordingly, unite the two advan- tages of being regulated by the sun, and of being perfectly uniform. The mean or average day of 24 hours must therefore be an average taken of all the days in the year, that is, such a day as the sun would regulate if he moved uniformly in R.A. This average day is called * In the questions which this figure illustrates, motion round the pole only is ronsi- Jf red ; sinre. therefore, the place of a celestial body on its meridian is unconnected with the motion of the meridian itself round the pole, no regard is had to declination. As th; spectator will naturally refer the hour-angle of a star to the elevated pole of the iilhce, in south latitude the figure will appear reversed, since the diurnal motion there appenm from right to left in facing the equator. The figure, however, may he drawn in th.it iiiannsil which may appear the clearest, the only point essential to be kept in view, bcnng that tht H.A is reckousd the opposite way to the ajiparent diurnal motion. 174 NAUTICAL ASTRONOMY. the mean sohir day, ami time thus regulated is called mean solar time, ur Mean Time, which is that shewn by clocks and watches. 476. The sun being generally either behind or in advance of tba position which he would have occupied if he had moved uniformly, mean time is in general either fast or slow, on apparent time. The correction for this irregnlarity, that is, the difference between tiie sun-dial and the mean solar clock, is eaikd the Equation of Time. Mean time is, therefore, deduced from apparent time, by applying the equation of time. See the Nautical Almanac, p. I. or 11., or Table 62. 477. The Sidereal Time at Mean Noon is the right ascension of the meridian at the instant when the sun, if he moved uniformly, would be on it. It is evident that this element, from its nature, varies uniformly ; iiow, since the sun's R.A. varies irregularly, and since the equation of time, which is the correction that removes this irregularity, must also vary irregularly, it follows that the unequal variations of the equation of time and the sun's R.A. are together equivalent to the single and uniform variation of the sid. time at mean noon ; and herein consists tlie great convenience of employing the sidereal time at mean noon, which has been given in tlie Nautical Almanac only since 1834.* 478. (1.) Let be the place of the sun, at about 4 P.M., m the place where he would be if he always moved uniformly ; then ©M is apparent time (No. 474), m M is mean time, and w is the equa- tion of time. The equation is here ad- ditive to a pp. time, as is the case from January to March, and from July to August. (See Table 62. ) (2.) Let T be the first point of Aries ; then, while the sun and t revolve, the sun moves contrary to the diurnal rota- tion, or is always increasing his R.A., or the arc -vN©, by nearlv 1° a-day. The complete revolution of -r constitutes a. sidereal day ; that of ©, an apparent solar day; and that of m, a mean solar day. After 24 sidereal hours the sun has still to describe about 1°, or Hie 360th part of the circle to complete it; the time necessary for which is about one 360th of 24 sidereal hours, or 4 sidereal minutes. Thus the solar day is longer than the sidereal day by about 4"". The mean solar day being divided into 24 hours, lh<; sidereal day is '23'' Se™ 4' of such a day. (3.) When m is on the meridian at M, the a.c M/n t, or the; * This element, which is the R.A. of a mean, or imaginary sun, is a very diffcent thinsr from the R.A. of the sun at mean noon, with which it has been confounded': the latter o»n differ only a few seconds from the R.A. at apparent noon, but may differ from the Sidereal Ti-ne at mean noon by the wliole amonnt of the equation of time, or si.vteen mi;iutes. DEPtNITIONS. 175 SiMi'fl mean R.A., is the sidereal time at mean noon. Wlien m lias arrived at m in tiie figure, tiiis quantity has clianged by an ann.unt proportional to tlie mean time M m. The © moves sometimes more quickly, at others more slowly , the ))oint m (which is merely an imaginary situation of 0, deduced by calculation, from knowing the limits within wliicli tiie irregu- larities of its motion are confined) moves equably. Hence m 0, the diiFerence of these two, changes unequally. (4 / By No. 472 (3) the sidereal time, or place of the point <r, is obtained from the hour-angle of any celestial body. By aj)plying to the place of t the sid. time at mean noon, we obtain the place of m, or mean time. Thus Mean Time is found from the hour-angle of a star. 479. Since the sun m {)asses over 15° of the circle in one mean :iour, he arrives at the meridian of a place 15° west of N M one hour after he has passed NM, that is, at one o'clock of the time at any place, or all places, of which N M is the meridian. In like manner lie jiasses a meridian 15° east of M one hour before he arrives at M, that is, when the time on M is 11 o'clock in the forenoon, or 23 hours after the noon of the day before. Thus the beginning of the day, and therefore the hour or time of the day, at one place differs from that of another place by the dif- ference of longitude of the places; the time at the easternmost of the two being in advance of, that is, greater than, the time at the other. Hence when tlie times proper to two places at the same instant are known, their diff. long, is determined, or the relative positions of their meridians.* 480. The Civil Day is dated from midnight, and the twelve hours are computed twice over; the Astronomical D^y is dated from noon, and runs through the twenty-four hours. Ei. 1. October 3d, %^ iS" p.m., civil time, is the same astronomical time. Ex. 2. January 3d, 4'' 25"" a.m. civil time, is reckoned January 2d, \b^ 25'° astronomical Et. 3. April 1st, II A.M. is, astronomically, Maicli 31st, 13 hours. 481. The Greenwich Date is the time at Greenwich corre- sponding to any given time elsewhere. f • The diff. long, is found as well by means of the motion of a star as of the sun, that is, by means of a clock or chronometer regulated to sidereal time, as well as by one regulated to mean time. For although the absolute interval of time employed by a star in moving from one meridian to the other is less than that employed by the sun, yet it is divided into the same number of hours, minutes, and seconds, but which are of smaller magnitude and thug (Jje difference of time results, in numbers, the same. t Here terminates all requisite description of the terms used in the rales in the present volume. The other terms which occur in the Nautical Almanac will be described in the TTifory. In this chapter we have sometimes spoken of the earth as fixed and the heavens ai movable, although this is contrary to fact, because the appearances alone furnish us with the measures of time, without any regard to the actual state of things. Again, we have considered the earth as a sphere instead of a spheroid (No. 180). The nnseqaeaces of the oblateness, in an astronomical point of view, are that the planes of ti* I 7Ci NAUTIUAI, ASTRONOMY. 482. It will be found a useful exercise of what has preceded to verify the following remarks: — (1.) No star of which the pol. dist. is less than the lat. can set, and no star of which the ])ol. dist. exceeds 90° plus the colat. (SM, fig. p. 162) can be visible. (2.) When tlie pol. dist. is less than the lat. the star passes the meridian both above and below the pole. (3.) When the pol. dist. is less than the colat. the star passes the meridian between tiie zenith and the pole, and does not pass the prime vertical. (4.) When the declin. is 0, or the pol. dist. 90°, the body rises and sets in the E. and W^. points. The hour-angle at rising and setting is (>'', and the body is seen raised on the prime vertical by tlio effect of refraction ; unless it is the moon, which, from her parallax being greater than her refraction, is not seen at the precise time of her rising and setting. The object is above the horizon for 12 hours, and 12 hours below it. In this case the amplitude is 0, except from the effect of refraction. (5.) When the pol. dist. exceeds 90°, the celestial body rises and sets on that side of the E. and W. points which is farthest from the elevated pole ; the hour-angle at rising and setting is less than 6" : the time during which the body is above the horizon is less than 12 hours, while it is more than 12 hours below the horizon. The body does not pass the prime vertical above the horizon ; and the amplitude is i-eckoned towards the S. in N. lat., and towards the N. in S. lat. (6.) When the pol. dist. is less than 90°, the celestial body rises and sets on the same side of the E. and W. points as the elevated pole ; the hour-angle at rising and setting is greater than 6''. Tiie body is more than 12 hours above the horizon, and less than 12 hours below it. The amplitude is reckoned towards the N. in N. Lat., and towards the S. in S. Lat. ; the body passes the prime vertical twice The hour-angle at the passage of the prime vertical is less than 6^. (See Table 29.) (7.) A star having a certain declination always rises and sets in the same points, and passes the meridian and prime vertical, or any other circle of altitude at the same altitude, without regard to its R.A. rircles of altitude (excepting the meridian) do not pass through the centre, and that the length of the radius, or line drawn from the centre to the place of the observer, is different in different latitudes. The first of these conditions produces no sensible effect in practice, because the Time is not affected by it, and the same Latitude (though differing from the latitude on a sphere by the quantity in Table 52) results alike from all observations, of whatever kind, of B body not affected by parallax, — and thus the oblateness, however great, would always be neglected in determining a place by observation of the stars or the sun. By the second condition the parallax of the moon is affected, and a further correction of her apparent place becomes necessary. We have also described the first point of T as fixed, whereas it has a very slow motiosi. The stars, also, though called fixed, have slow proper motions. These and other points twt Me«'c»sary to our present suliject will be treated more at large in the Theory. DEFINITIONS. 177 (8.) As tlie piece of a star or ;iny celostial body is dotcrnilnod by its 11. A. and Decl., and as, at the jdace of the spectator, the peti- tion of the celestial equator, to which both these are referred, id fixed, it is easy to know whereabout any star is to be looked for at any time. When, as is couinionly the ciise, the time (mean or appa- rent) is given, the sun's hour-angle is known ; and therefore, when he is invisible, his place on the equator may be estimated. By means of the sun's nlace, and his 11. A., the place of the first point of Aries may be estimated; then the star's R. A. gives the place of its meri- dian on the equator, and its declination the place of tbe star with respect to the equator. When the sidereal time is given, the place of the first point of <r is at once known, just as the place of the sun is known from the apparent time.* • The position of the equator, and the relations among the Latitude of the place, the Time, nnU the Hour-angle, Altitude, and Azimuth of a celestial body, are best illustrate.l by a celestial glolie. The liroad horizontal rim represents the Kational Horizon (No. 421). The brass meridian of the globe being laid N. and S., and the Pole elevated, by the dcgries marked on it, to the latitude (No. 449), ihe globe represents tbe celestial sphere as shewn in tigs. 1, 2, p. 162. The position of the sun is found by marking the sun in his place in R.A. and Decl., by the help of tbe divisions on the globe, and then setting the sun at his proper hour- angle by means of the hour-circle near the pole. The Alt. or Zen. Dist. is measured by a graduated slip of brass, or by a thread, as in the note, p. 129. It is un- necessary to enttT further into details, ns the reader wbo well understands the definitions aliove will find no difficulty in solving any useful " problem on the globe " which can bo p;c>pose<2, without burdening bis memory with technical rules. In the absence of a globe, distinct ideas may be obtained of the actual positions of the celestial bodies by a circular card, as a compass-card, having the liours marked on the edge, and an axis, as a pencil, put through the centre perpendicular to the card. If this a.\is be laid N. and S., and the north end (in north lat.) raised up till it is inclined to the horizon at an angle equal to the latitude, it will represent the pol.-ir axis round which the celestial bodies revolve, the card representing the equator. The C' being brought up to the meridian, the hour of the day at the edge will shew the place of the sun's meridian at tho time. If the C' be made the first point of T, the hours become hours of R.A. ; if, then, the © be marked on the edge, on its proper R.A., and then turned round to the position proper to the hour of the day, the place of tbe first point of T is seen. Suppose, now, a small telescope were placed on the axis making an angle with tho plane of the equator, or the card, equal to the declination of some star, then, while this elar revolves parallel to the equator, the telescope, kept at the sime angle, could at any time be directed to wirds the star by merely turning the axis round, \ large instrument i6 <waEtruct«d as tiut piinnple, ati is sailed en EquatorcaU CHAPTER 11. Instruments of Nautical Asthonomt. I. The Eeflecting Instruments. II. The Artificial Horizon. III. The Chronometer. I. The Reflectixg Instruments. 483. These are instruments for measuring angles between two objects, by bringing the reflected image of one of them to coincide with the other seen directly. They are necessary for observing altitudes of the heavenly bodies at sea, where the spectator has no fixed point of reference except in the horizon. On shore, and often on a field of ice, the fixed point required in observing alti- tudes is obtained by means of the artificial horizon. 484. The instruments of this class which are in most common use are the quadrant, sextant, and reflecting-circle. For conve- nience, we shall describe the adjustments generally under the two former; and as every person in possession of an instrument will be instructed by the maker or some expert person in the names of the different parts, and also in the mode of handling it, and packing it in the case without danger of distortion, we shall confine ourselves merely to matters of general reference. 1. T/te Quadrant and Sextant. 485. The quadrant contains an arc of more than 45°, and measures a few degrees more than 90° ; it is usually made of wood, and the graduated arc, which is ivory, reads to minutes, and sometimes to 30". The sextant measures a few degrees more than 120° ; it is made of brass, and sometimes reads to 10". The quadrant serves for common purposes at sea, but the sextant is required for taking a lunar observation. The observer should be in the habit of employing good instru- ments of their kind, as inferior instriiments naturally induce careless and imperfect observation. 486. The sextant made of a very small size, and thence called the Pocket Sextant, is adapted to the use of surveyors, tra- vellers and others, on occasions in which minut<} accuracy is not uecessary. INSTRUMKXTS OF NAVTICAI, ASTKONOMT. 170 [1.] Marnier of tlnug. 487. To take tlie sun's altitude at sea. Set t.lie index at 0, put down a screen before tlie central mirror, hold the instrument in a vertical position, and direct the sij^ht, through the sight-vane anil liorizon -glass, lo that part of the horizon which is exactly under the sun. Now move the index on with the left hand, and the image of tlie sun will ap])ear to descend towards the horizon. Vibrate the instrument round the line of sight, and make the lower limb touch the horizon : this gives the observed altitude of the lower limb. 488. .This last altitude is sometimes near enough ; but for accu- racy, having made a rough contact as above, put in the telescope, previously set to distinct vision by looking through it at the liorizon-, the image being now magnified, the contact is made more correctly. In general the telescope should not be fixed till a I'ough contact has been made, because it narrows the field of view, and increases the difficidty of bringing the images together. The contact must be made in tlie centre of the field : if it is too near the plane of the instrument, or too far from it, the angle will be too great by the quantity in Table 54.* 489. When there is a tangent-screw, clamp the index, and make the contact perfect by turning the screw, — some further remarks on which will be made in the ])roper places. The tangent-screw should be kept nearly middled when not in use. 490. To take the altitude of a star. Set the index to 0, direct the sight to the star, hold the instrument vertically, and move the index onwards; the image of the star will be seen to descend. This method is proper to avoid bringing down the wrong star, but should not be practised with the sun, as it exposes the eye to an intense light, which may derange it ibr the whole observation. 491. The shades, or coloured glasses, placed before the two mirrors, tend to equalise the brightness of the object and the image, and sometimes distinguish one from the other by the diflference of colour. The shades require to be particularly well ground, because, if the surfaces are not strictly parallel, the rays in passing through tiie glass are turned out of their fornier direction: hence, when a defective shade is placed before each of the mirrors, the angle is aflected by the sum or the difference of the errors due to the shades. It is advisable, therefore, in general, to employ a dark glass at the eye-end of the telescope, by which the shade before one or both of the mirrors maybe dispensed with. Also, if this glass is not per- fect, the rays from the object and the image are affected alike, and the angle between them remains unchanged. A card screen, to sli|) over the eye-end of the telescope, is useful in protecting the eye from accidental glare. 492. The observer acquires, by attention, the power of estimating * Mr. Hartniip, director of the obsenatory at liivcrpool, act|uaints me. that he has con- gxv.'.\y found sextant observations to come out more accurately in proportion as he narrow J tt« field 1>y closing the wires. 1^0 NAUTICAL ASTRONOMY. the proper angle at which to set the index for a rough contact, and tiius saves time. It also effects some saving of time to liave the tubes of the telescope marked at the observer's focus. 493. When the angular distance between two objects is to be measured, the plane of the instrument is held in the line joining them, and the sight is directed to the fainter of the two. When, therefore, the brighter object is to the right, the instrument is held face upwards, and the image of the right-hand object brought to touch the left-hand object seen directly ; but when the brighter object is to the left (as in observing the distance between the sun and moon in high north latitudes in the forenoon), the instrument must be held face downwards, the sight being directed to the right- hand object. The contact must be made in the centre of the field, as directed above. [2.] Reading off the Angle. 494. The angle having been observed, its measure is to be read off. The arc being divided into degrees, and these subdivided into halves, thirds, &c., the smallest division contains several minutes, and the angle can thus be read, but roughly, from the arc itself. In order to read to 1', or a fraction of 1', a scale called a vernier is applied to the arc ; this is a portion of an arc having the same centre, and divided into one part more than an equal portion of the arc itself. The manner in which a more minute reading is obtained may easily be understood from the following example : — Suppose a division on the arc to be ^ of 1°, or 20', and the vernier to be equal in length to 19 divisions, or 6° 20', but divided into 20 equal parts ; then each of the divisions on the vernier is Jj of 6*^20' or 380^, that is 19', and therefore the difference between one division on the arc and one on the vernier is 1'. Suppose the beginning of the vernier and that of the arc to coincide, as in Fig. 1 ; then the first of the dividing lines of the vernier falls short of the first dividing line of the arc by i'; therefore, if we make these lines coincide, we advance the vernier i . Again, to make l^e second dividing lines of each coincide, we must move the vernier through 2', and so on. In Fig. 2 the o of the vernier stands between 20' and 40' after the division at 5", and the iil-st coincidence is at 9 ; hence the arc measured is 3° 29'. ng. 3. INSTRUMENTS OV NAUTICAL ASTRONOMY. 181 When the iiiJex is moved the contrary way, the o of the vernier goes off the arc, af seen in Ktg. 3. As the 20 of the vernier stands at 6° xo' when the two zeros coincide, if we move it i' to the riglit, the coinddjnce will occur at 19, and at 18 if we move it 2', and so cii. Hence, to measure an angle off the arc, we must read from the end of the vernier. The an shewn is 32' off the arc [3.] Adjustments. 49r). ({.') Tlie Index-Glass, or central mirror, must be perpeB- (licular to the platie of the instrument. Set the index about 60°; then, if the image of the arc in the mirror appear in perfect continuation with the arc itself, the adjust- ment is perfect; if the reflection seem to droop from the arc itsoif, tlie mirror leans back ; if it rise upward, the mirror leans forward. The position is rectified (in quadrants only) by the screws on tlie iiack. This adjustment generally rests with tlie maker, but it should be occasionally verified by the observer. (2.) The Horizon-Glass, or fixed mirror, must be perpendicular to the plane of the instrument. Set the index to 0, hold the instrument horizontally, look through the glass at the sea-horizon, or other distant object, and give the instrument a small nodding motion : then if the reflected image appear neither above nor below tho real object, the adjustment is perfect ; if the image be the lower, the glass stoops forward ; if it be the hit/her, the glass leans backward. The position is rectified by the screws. (3.) The line of sight of the telescope must be parallel to the |)lane of the instrument in which the index moves. Place the two wires of the telescope parallel to the plane of the instrument. Select two distant objects from 100° to 120° apart, as two stars, or the sun and moon, and make an exact contact at the lower wire, or that nearest the instrument. Now move the instru- ment so as to throw the images in contact upon the upper wire; if the contact is still perfect (the images having overlapped in the middle of the field), tlie adjustment is perfect ; if they have separated, the object-end of tlie telescope droops ; if they overlap, it rises. The position is rectified by the screws in the collar. When this adjust- ment is defective, the observed angle is always too (jreat. (See Table 54.) [4.] Index.P.nor. 496. The graduation of tne arc should commence at a certain point ; when this is not the ease, the Index-Error, as it is called, must be measured. The point at which the graduation of the arc is sujiposed to begin, is that at which the index stands when the mirrors are parallel, as is tiie case when the image of a distant object is seen to coincide with the object itself. The index-error, therefore, is merely the error of the place of the beginning of the divisions, and afiects all angles alike. To find the Index-]'2rror. (1.) By the Horizon Hold the instru- ment vertically, and make the image of the horizon coincide with the boriion itself as accurately as possible. If the 0, or zero of the Ex. 2. On fue arc 30' 10" Otf 33 40 Ind. Corr. add 3 30 • 45 IS2 NAUTICAL ASTRONOMY index, now stand at 0, there is no index-enor ; if it stand on the arc, tlie index-correction is so much subtracdve ; when o^tlie arc, additive.* Ex. The horizon and its image being maiie to coincide, the reading i.'? %' Oil the arc Then 3' U the Index Correction to be subtracted from every angle observed. Any distant object, or a brijjht star, answers the purpose. (2.) By the Sun. Measure the sun's horizontal diameter, f moving the index forward on the divisions; read off the measure which will 1)6 on the arc ; then cause the images to change sides by moving the index back; take the measure again, and read off; this reading will be off the arc: lialf the dilFerence of the two readings IS the index-correction. When the diameter on the arc is the greater, the correction is subtrnctive ; when the ksser, additivc.X Ex.1. On the arc 32' 10" j Off 29 5 !no. Corr. subtract i 10 1 In consequence of the spring or elasticity of the index-bar, the L-rror will be different for the onward and for the backward motion of tlie index. It has been recommended, tlierefore, to turn the tangent-screw right and left alternately, in making successive con- tacts, by which a partial compensation is obtained. This source of discrepancy is, however, effectually removed by taking all observa- tions, including tliat for index-error, with the same motion of the index-bar. The ontvard motion being adopted as the most natural, tlie tangent-screw is always employed to close the object and the reflected image, and is thus always turned in tlie same direction.^ One-fourth of the sum of the two readings should be equal to the sun's semi-diameter in the Nautical Almanac. Tiiis affords a test ot tlie accuracy with which the observation has been made. 497. Tiie adjusting screws are uefer to be touched except from * When the mirrors are parallel, a very distant object is exactly covered by its image ; but at a near object the distance between the mirrors subtends a sensible angle, or has sensible parallax, and this coincidence does not take place. The parallax of a 12-inch sextant St h.-Jf a mile distance is about 21", and is smaller for smaller dimensions and greater dis- tances, in simple proportion. Hence, for the purposes of adjustment, distances exceeding this should be employed. Captain Beechey suggests a method of adjustment by parallel rays. Naut. Mag. 1844. p. 505. f As the refraction increases towards the horizon, the lower limb is more raised than the upper limb, and the vertical diameter is shortened. This, at very low altitudes, produces d flattened or oval form in the sun and moon. X If both readings are on the arc, which can only occur when the index-error is nearly half a degree, the ind. corr. is the mean, and subtractive ; if off, additive. § Sir F. Beaufort, to whom I am indebted for the suggestion, acquaints me, that from the sensible influence of the spring of the index-bar in nice observation he ■uniformly adhered to this plan, and caused it to be followed by his officers. The late Captain Basil Hall informed me that he made it his practice to obtain the index-error both for the onward and the backward motion of the index employing the former error in all observations by the onward mntiou, such as the lunar disfarce wh«"n Increasing, and the latter in observations by tlie reverse motion, as for the Innar distsooe nberi decreasing. lylSTRUMENTS OF NAUTICAL ASTRONOMY. 183 necessiii/, and then with the greatest possible caution.* When two screws work against eacli otlier, care must be taken, in tightening one, to loosen tlie other if necessary. 498 Besides errors from these causes, there are others which are neither detected nor remedied so easily : the divisions on the art; are liable (though in these days in a very sliglit degree) to inaccu- racy, and the centering of the arc is not always perfect. f In order to test the accuracy of the arc in either of these respects, in different places, it has been proposed to measure the distance of two stars, comparing the distance with that shewn by a circle, or by an approved sextant, or deduced from calculation. J Tiie absolute error being thus found for certain places on the arc, the correction for any angle may be inferred by proportion. 499. As the two sides of the coloured glasses are not always exactly parallel, the shades may vitiate the angle. (No. 491.) Some observers find, by actual trial, the error due to any shade or com- bination of shades. Tlie shade in the eye-piece, as before stated, has not this defect ;§ but an image-shade is generally indis])ensable in taking a lunar observation. [5 J Methods of Increasing the Efficiency of the Stxtant. 500. Tlie necessity, under certain circumstances, of observing large angles, and the difficulty of measuring them, arising from the obliquity with which the rays of light, in such cases, fall on the central mirror, have led to the suggestion of various plans for extending tiie powers of the sextant. Capt. Fitzroy has employed an additional fixed horizon-glass, placed at a constant angle with the ordinary one, by Jiieans of which the image of an oiiject above, or to the right-hand of another in the * Particular attention is called to t'ciis point, because it is a common failing of " over- handy gentlemen" (to use Troughton's language) to "torment" their instruments. It is better that error should exist, provided that it is allowed for nearly, than that mischief should ensue to the instrument from ignorant attempts at a perfect adjustment ; and the skilful obser^'e^, instead of implicitly depending upon the supposed perfection of his instrument, wil! endeavour to avail himself of those cases in which errors, if they e.xist, will destroy each other. t It is also necessary that the two surfaces of the central mirror should be exactly parallel. This paralleli.sm can be tested only by observing an angle between two objects i2o'^ or 130° apart, and then repeating the observation with the mirror in a reversed position. Half the difference, if there is any, between the two results is the angle between the sur- faces. As in the best instruments the mirror is fixed, this cannot be put in practice, and the consideration is therefore omitted from the adjustments in the text. This error, how. ever, when it exists, is obviated by the method described in the next sentence of the text. t The stars for this purpose must be taken from the Nautical Almanac, as the places are required with precision. The true distance may then be computed by the rule No. 339 (2), using the Diff. of the stars' right ascensions for D. Long., and their polar distances for the colatiludes. The true distance may then be reduced to the apparent (which is that mea- sured by the instrument), by No. 842, substituting one of the stars for the moon, omitting the second corr., and applying the other star's correction the opposite way to that laid down in the tabulated directions for the star. § Working with the artificial horizon, the eye-piece of the Inverting tube should, if po.'Bible, be used insiea<l of the shades of thu sextant ; if shades are used, endeavour always to use the same. The meridian altitude of the sun should, if possible, be observed with the eye-piece, as the latitude olilaliied from it can then be aieaned mora Kitislaclorily with that determined by the stars. IS4 NAUTICAL ASTROXOMY. line of sight, is seen in the field when the index is at 0. and thus a portion of the angle is measured in addition to that on the arc. ■501. Admiral Beechey had a sextant constructed with a second central mirror over the usual one, and working on the same pivot, the arc of which, being concentric with the usual arc, is divided l>y the same stroke. Both index-glasses are adapted to the same liorizon-glass.* Any angle is measured by putting one index forward upon the arc to any convenient number of degrees, and moving the other until both reflected images are seen in the horizon-glass. Each arc has its proper index-error. 502. Mr. C. George, R.N., has constructed a double pocket- eextant, by joining two small sextants by the face. This instru- ment, which scarcely exceeds the box-sextant in size, possesses for various approximate purposes, and for surveying, the advantages of the double sextant. t o03. The double sextant has some important advantages; it affords two alts, of the same or different celestial bodies in quick succession: this is a point of much consequence when the body appears for short intervals only, as between flying clouds, and also in observing at night, as it saves the disturbance to the eye caused by reading off; it measures the angular distance between opposite points of the horizon, J and thus serves as a dip sector ; it measures two terrestrial angles at the same instant, and thus serves as a director. The index-error of a compound angle measured by a double sextant is composed of the errors proper to each are. The error of parallelism (No. 495) in a compound angle is mate- rially reduced, since in practice each portion is less than 90°. 504. In observing altitudes at sea by the double sextant, set any angle on the upper sextant ; then, facing that part of the horizon nhich is opposite tlie sun, find his image, and bring up the horizon to the lower limb; by moving the lower index : the sum of the two readings is the suppl. of the alt. of the upper limb, affected by the dip and the index-error. Now unclamp the indexes, set the upper one to an angle less than the alt., find the image under the sun, and bring up the horizon to the lower limb: the sum of the readings is the alt. of the lower limb, affected by the dip and the index-error. Half the difference of the two sums is the app. zen. dist. cleared of the dip, semi-diameter, and index-error. * Admiral Beechey acquainted me that he constructed this sextant for the purpose of obtaining the miasures of the unfiles between two tenestiial objects at the same instant anil by one observer : a point of considerable importance in surveying, or in l'iyinf;do«n soundings, while the observer himself is in motion. A luriher advantage afforded by the ^instruction is, that when the right-hand obj-ct is too faint to be rifleeted, the sexttnl does not require to be inverted. The instrument is eonsiiucted by Gary. t Made by Gary. I The ditfcrenee between this angle and iSo" is twice the apparent dip. Thus, if this nngle, measured downwards, is 179° 48' 30", the apparent or actual dip is 5' 45". The dip sector, being mconvcuient and little used, is not described in the text. INSTRUMENTS UK NAUTICAI, ASTIJONOMV. 185 2. The litpeutini/ Rcfeclni<j Circle. 505. On this circle the measure of the angle observed by reflec- tion, as in a sextant, is carried over any part or tlie whole of the circumference : this is effected by niaking the horizon-glass itself movable round tiie centre, and attaching to it a vernier. By tiius repeating tlie same angle on different parts of the divided edge, the errors of the index, of the coloured shades, and of the centering, are nearly, if not altogether, removed ; also, since the indexes follow each other round the circle (each mirror alternately acting the part of the fixed horizon-glass), the angle finally registered is the sum total of all the re]ietitions ; and thus one reading alone contains tlic result of any number, however great, of separate observations. The arc read off, divided by the number of observations, gives the mea- sure of tlie required angle. 506. When the angle changes during the observation, tlie arc finally registered is not the mere repetition of the same angle, but the sum total oi different angles ; it is therefore necessary to under- stand how the time is to be noted. Suppose, for example, at 5'' 20"' the angle is 45°, and at 5'' 16"' it is 46° (neither being read off) ; now, at 5'' 20"' the first index would shew 45°, and at 5'' 26'" the second index would shew the sum of 45° and 46°, or 91°, halfoi which, or 45° 30', in this case obviously corresponds to the middle time, 5'' 23"'. The same appears generally thus : the last arc read off measures the first angle, the repetition of the same angle, and the change upon it during the interval of the two observations; therefore half the arc measures the angle, and half the change upon it, supposed uniform, Mhich corresponds to the middle time. If, now, a second pair of angles, as before, be observed, a second angle with its time is obtained, and so on ; hence, as long as the change of the angle is uniform, the arc read off, being divided by thfl number of observations, corresponds accurately to the mean of the times. The time is therefore to be noted at each contact. 507. The Circle is made in various forms: we sliall confine our- selves here to the description and use of those known by the names of Borda's and Dollond's Circles.* Figures are purposely omitted, and the general description will be easily followed with the instru- ment itself. In using ttie circle, care must be taken to push tne crooked handle out of the way of the telescope. * Troughton's Reflecting Circle, which docs not rejjeat, is capable of great precision ; but it does not seem so well adapted to general practice, especially at sea, as the repeating circle t for the three indexes aggravate the inconvenience and tediousness of reading off ; and the in.strument, instead of facilitating, like the repeating circle, the multiplication of observatior,;?, itfords merely a correct mesi'iure of an angle which, from the motion of the ehip, i< itself cb«r;ed icAccu/etcly. 186 NAUTICAL ASTRONOMY. [1.] Borda's Circle. 508. In Borda's Circle, the horizon-glass and telescope revolve together round the centre, like the central mirror, carrying a vernier, which we shall call A. Sometimes another vernier is placed opposite to A, and moves with it. The central mirror carries, like a sextant, a vernier, which we shall call B. The circle is divided into 720°. The Iiorizon-glass and telescope are attached to an inner circular ivrc divided to degrees, wiiich is called the finder, as it enables tiie mirrors to be set to contain any angle, and the objects can thus be at once brought into contact roughly. When B is set to at the middle of the finder, the mirrors are parallel. The divisions on the finder are reckoned in both directions from the 0. 509. To use the circle as a sextant. Before this can be done we must know the reading of B when the mirrors are paralhd. To find this, set A accurately to 720",* and clamp it. Set B to on the finder, nearly, and measure the sun's horizontal diameter : read off. Cross the reflected image to the other side of the sun, and read oft': the mean of the two readings is the constant angle required, and is clear of index-error. To observe, move B as in a sextant. After observation, examine the setting of A, as any error in this IS so much index-error. 510. By moving the index opposite ways, observations may be taken backwards and forwards, from the same point on the arc ; but tlie real efiiciency of the repeating circle consists in what is called the cross-observation, to which we shall now proceed. To observe an Altitude by the cross-observation. Set At accu- rately at 720° (or at 360°); set B to on the finder roughly ; observe the alt. with B as with a sextant; read oft' B roughly on the finder; unclamp A, and move it on the finder, in the order of the divisions on the circle, till the on the other side of B stands at the angle read oft'. Turn the circle over, hold it in the other hand, and com- plete the contact by turning the tangent-screw of A. The vernier A now registers the first pair, or double the altitude required. To proceed with the repetition. Unclamp B, set it on the finder at the same angle as before; hold the instrument as for the first observation ; complete the contact. Unclamp A, move it onwards as before till the stands at the angle read off; complete the con- tact. This is the second pair, or four times the required altitude. * Tnis index will, in some circles, stand at 360°, and may require to be moved back- wards ; 360° would then be subtracted from every angle measured by this index alone. The above instructions will, with a trial or two, be found sufficiently intelligible. t It is usual to fix first the index called here B, as directed by Borda himself, and repeated by other writers ; but it is immaterial which index is first fixed, or at what part o< the circle, provided tlie vernier be read off. The index A is recommended here in order to ivriiiimiUte as much as possible the use of trie circle to that of tlie instruments with wliich •.VI! are already more familiar, luaccuracy ia this setting is iliminished as tlie number oi le|ietitions is increaj-:!!. rXSTUUMENTS OF NAUTICAL ASTRONOMY. 187 riic next reading of A will be six times the required altitude, aatf iO oil. 511. To observe Angular Distance by the cross-observatlci:. Proceed as directed above, reading distance for altitude. 512. If tliere is not light enougii to read tlie finder, the reflected image must be actually carried across tiie other object by movi:;:; the index tlirongii twice the angle first measured. 513. The last [lair completed being registered by tlie vernier -A, the disturbing of 13 at any time is immaterial, since it does not att'ect the reading of A ; but if A is moved, and tlie observation is inter- rupted before tlie new pair is com])leted, the whole is lost. 514. Two altitudes, of the same or different bodies, may bo obtained by reading both verniers;* thus, set A to 720", observe one alt. with B, as in No. 509. Unclainp A, move it to on the finder, hold the circle in the other hand, and observe the other altitude. Read off B, and sul)tract from it the constant angle : tlie re-i mainder is tlie first alf,. For the second alt. subtract the first alt. from A. Ex. 3 251' 2'; A 98" 11'; const. 213° 35'. The Fmsr Alt. is 3S' 27'; the Second is 59" 44'. 515. We shall now consider the effects of errors. The index- error is obviously removed by measuring tlie same angle, either on opposite sides of a fixed zero, or between any two points on the arc. Now, after B has been clamped, and the angle is to be repeated by moving A, the horizon-glass passes from one side of the perpen- dicular upon the central mirror through the same angle on the other side; the angle, therefore, is meiisured by the motion of A from one point of the arc to another, and the exact point 720° is assumed merely for convenience in reading. When a coloured shade is defective, it breaks the direct course of tlie ray from the central mirror to the horizon-glass, and the broken part inclines towards the same side of tlie horizon-glass, wiiether the circle is inverted or not. Therefore, if the angle formed on one side of the perpendicular on the fixed mirror is too great, the angle formed on the other side will be too 'mall, by the Bame quantity, and this error disappears. 'Ihe inclination of the line of sight upon the plane of the circle. No. 495 (3), produces the same effect upon the angle formed u])on eitiier side of the jjerpendicular to the central mirror; this error therefore remains. The error of the eye, and therefore the personal equation (No. 17.5), likewise remains. The error of centering is removed by carrying the angle round the whole circumference. * This may be found convenient in taking a lunnr at nii;ht, since tlie lamp would be required hut three times for reading, in obtaining the four altitudes required and the suver.O pairs of distances. Rules might easily be given for repeating both altitudes to any ext.'.sit, «Kit an allowance would be necessary for the motion in altituJc of the second body obsWtfrL' 188 NAUTICAL ASTRONOMY. [2.] Dollon<Ts Circle. 616. Dollonu's Circle consists of two concentric circles, the inner OJie of which, in revolving within tlie other, carries the horizon glass and telescope, and a vernier called A, of which the clamp and tangent screw are attached near the telescope. The inner circle is cut to degrees only ; the central mirror carries a vernier called B, as in a sextant. The inner circle answers the purpose of the finder above de- scribed. From the position of the telescope, this circle is held, in taking altitudes, exactly like a sextant, which is a convenience. From the general resemblance between the two instruments, it is unnecessary to enter into further details.* II. I'he Artificial Horizon. 517. The Artificial Horizon is a small shallow trougli, a few inches in length, containing quicksilver or any other fluid, the surface of which affords a reflected image of a celestial body. The fluid is protected from the disturbing efl'ects of the air by a roof, of which the two opposite sides contain plate-glass. This roof is often made to fold np for the sake of portability. The trough should be 90 thick as to raise the quicksilver to a level with the lower edges of the glasses. A piece of talc, which substance splits into thin parallel plates, may be laid on the trough as a substitute for the roof. In some cases a piece of thin cloth, as muslin, sufficiently transparent to allow a bright object to be seen through it, protects the fluid from the wind. 518. The image of a celestial object reflected from tli3 surface of a fluid at rest appears as much below the true horizontal line as the object itself appears above it ; the angular distance- measured between the object and its image is therefore double the altitude. An advantage resulting from this is that in halving the angle shewn by the instrument we halve, at the same time, all the errors of ob- servation. The reflected image in the fluid is always less bright than the object, but as it is perfectly formed, and as the surface is truly horizontal, the artificial horizon, when it can be employed, is always to be preferred to the sea-horizon. * It is the opinion of some competent judges that circles should be made much smaller, for the sake of lightness and portability, and that they should accordingly be cut to minutes only, as Borda's Circle formerly was j because, by repetition, the minute or nearest half, minute read otF is speedily reduced to quantities smaller than can be measured in ths observation. The case of a sextant, or circle, should be made to receive the instrument permanently with the index in any position, as the reading off, which is always difhcult in defective light, Blight thus be deferred to a more favourable opportunity. It would also be useful for reference in cases of error or doubt in the reading, espe^'ially at night, to leave tlie iudej indisturbed till the result had been worked out. INSTRUMENTS OF NAUTICAL ASTRONOMV. 189 When trie uitifiule exceeds G0°, tlie altitude by rcfleelioii ox- iTeediiifr 120° falls without the iiiiiits of the sextant. In h>\v JJiti- tudes, therefore, it is often impossible to observe with the qiiiclisilver except by a sextant witii additional powers.* On tiie otiier hand, when the altitude is low, the observer is obliged to inc-rease iiia distance from the quicksilver, by wliich it becomes difficult to keep sight of the image reflected in the fluid; and for altitudes less tlian 12° or 15° the observation is generally impracticable. 519. The roof should generally be placed upon asheetofsorao thin material, impervious to vapour, wliicli, condensing on the glass, obscures tiie image. A leaden stand about the size of an octavo volume, on three legs, and covered with cloth, into which the roo/ Bmks and excludes the external air, is convenient. 520. The film, or scum, which forms on the quicksilver, is pre- vented from running into the trough by holding the bottle inverted while it is poured out. A wooden scraper, fitting close to the inner breadth of the trough, has been found to remove the scum, which adheres to the wood. 521. The fluid proper for the purposes must possess the qualities of giving a bright image, and of quickly subsiding to a perfect level efter being disturbed, such as quicksilver, wafer, spirit, and otliers. An ingenious, handy, and portable mercurial horizon by the late Captain George, R.N., made by Gary, 181 Strand, is recom- mended. It consists of a disc of glass floating on mercury, in a vessel which it nearly fits, and it has an arrangement by which the mercury is introduced, ready filtered from an attached reservoir, and afterwards withdrawn, in a manner which saves a great deal of trouble. The glass floats without touching the sides of the trough, and the whole of the mercury below is ser- viceable. Anot^her advantage is, that the edges of the trough cut ofif proportionally less of the field of view, hence very low altitudes may be observed with this instrument- The glass must necessarily be of the best workmanship. When the air is calm, a piece of water, or a puddle large enough merely to exhibit the image, is often a complete substitute for the quicksilver. t 522. As the celestial bodies are sometimes distinctly visible when the sea-horizon is enveloped in mist,;]; attem])ts have been made to * To remedy this defect, it has oeen proposed to use a reflecting surface, inclined at a coiut&nt angle to the horizon, movable on a level surface or floating in quicksilver. Also, a •extant has been tixed, with its plane vertical, to a pill.ir turning on an upright axis, and the telesi-ope laid nearly horizontal by a spirit-level, the image of the body being brought down to a horizontal wire in the telescope. + A small piece of plate-glass levelled by a bubble is sometimes used, but the performance of thi« instrument is not always satisfactory. t Capt. Scoresby (" Journal of a Voyage to the Northern Whale Fisliery," p. I.'i9), re- marks, that fogs often cover the sea in the polar regions to the depth only of 150 or 200 feet, while the sky is perfectly clear. Her Majesty's sloop Zebra was a week without interruption in a dense fog, to the south- ward of the Snares, during the whole of wliich tiir? no observation could be taken, though *.he sun often shone brightly (Naut. Mag. 1844 ). The like circumstances occur in "the Smokes," on the onast of Africa. 190 NAUTICAL ASTRONOMY. obtain an artificial horizon adapted to be used on board ship, by means of the siu-face of a viscid fluid, and a mirror attached to a pendulum, which, by its weight, hangs vertically.* The objections to the first of these have already been stated. With regard to the motion of a pendulum, it is important to observe that when the ship comes to the end of her roll or lurch, it does not »t once rest in the vertical position, but continues to move onwards, or to swing, witli the velocity which it had before the shiji's motion was destroyed ; hence the ]3enduluui moves through greater angles than the ship. By combining, however, the viscid fluid and the pendulum. Commander Becher has obtained a method of measuring altitudes at sea, independently of the horizon, whicu appears, from the reports made upon it, to afford sufficient accuracy for common purposes, when the motion of the ship is not very great. f Outside the horizon-glass of the sextant is a small pendulum, an inch and a half long, suspended in oil ; to this is attaciied a horizontal arm, carrying at the inner end a slip of metal, the upper edge of which, when seen in a certain position, is the true horizon. The error is determined by observation of a known altitude, or by the help of another sextant, and is the same for all altitudes. It should be frequently examined. A lamp is attaciied for observing at night. 523. Admiral Beechey fitted, within the telescope of the sex- tant, a balance carrying a glass vane, one half of which is coloured blue, to represent the sea-horizon, and to which the celestial object is brought down. The amount of oscillation above and below the level is indicated by divisions on the glass, tlie values of which are determined by the maker. The instructions for using this instrument are as follows:— Bring down the object, as the sun's limb, to the edge of the blue and leave it there. As the ship rolls, catch with the eye the upper and lower divisions reached by the oliject, and call them out to an assistant. who writes them down with the time against each. When two or more such readings have been taken, read off the alt. and write it down. Take the mean of the readings of the vane and turn it into src according to the scale furnished. When the mean is above, tlie e-ige, add it, when heloiu, subtract it. Apply the maker's index-error; tlio result is the apparent alt. being clear of dip. Ex. Took an alt., and readings as follows; the divisions u' each: — h ro s Divis. " ' lo 50 o (+1) above \ Observ. Alt. 20 25 20 50 30 I - , |) below ^ Mean of Div. -b 50 50 (+1.S) above \ ^ 10 19 20 51 20 (-2) below ; Maker's IiiJ. Corr . - 40 Mean 10 50 40 (-J), 2.5 above, ^J below; diff. i App. Alt. 20 iS 40 below I the half is ^ of 12' or 6' to be mb. • It has also been attempted, but without success, to employ the principle upon which a icp while spinning tends to preserve a vertical position, by balancing a horizontal mirror on 11 pivot, and causing it to revolve with great velocity. t See Naut. Mag. 1844, p. 291. Several reports, with observations made bv this iiisf.M- 'iisiit, will be found in the Naut. Mag. of 1839, 1842, 1844, &c. IN8TRUMF.NT8 OF NAUTICAI, ASTRONOMT. 101 Care is to be taken to observe as near tlie centre of tlie field a* possible, and exactly under the sun ; the elbow should rest on some tirm support. With practice tlie instrument affords considerable accuracy ; and in smooth water the mean of some alts, will be within 2'. A lamp illuminates the telescope at night.* 6"24. An instrument for this purpose, indispensable when the horizon cannot be seen, will also be of great service as a check, when haze or fog, by its partial distribution, produces the ajipear- ance of tlie horizon where it is not.f The same applies to tlie uncertainty in the place of the sea-horizon which is often expe- rienced in moonlight nights. These instruments are very convenient on shore. 111. The Chrono.meter. d25. Tiie chronometer is a superior kind of watch, furnished with an apparatus by which the changes in the rate arising from the expansion or contraction of the materials by heat and cold are nearly obviated. Chronometers should be kept near the centre of gravity of the ship, which is a little below tlie water-line, and not far from the middle of the lengtii, not so mucli because the motion liere is less than elsewhere, as because the temperature below is not liable to sudden changes. In ships in which great attention is paid to the chronometers, they are usually kept in a small apartment abaft the mainmast, on a table, in cases lined with cushions of soft wool, which defend them from the jerks and vibrations of the ship. The table is secured to a beam of the deck below, and in small vessels sometimes rests on a stanchion rising from the kelson. Large chronometeis are placed in jinibals, in order to preserve a horizontal position, as inclining a watch from this position affects its rate. They have also been hung, perhaps with the view of obtaining both these objects together, in swing trays ; but as this method is found to be very unfavourable, it has been discontinued. | The chronometer-table has been itself placed in jimbals. It has also been supported by springs to diminish still further the effect of shocks. o'26. When a chronometer is placed on board it should always remain in the same position, that is, with the XII towards the same * Made by Cary. t Adm. Bayfield acquaints me that he has been complutcly deceived in the place of the borijon at the coming on of a fog. i Mi. Fisher acquaints me that he has found vn acceleration of seven seconds a-day produced by suspending a chronometer in a cot with five inches' swing. 192 NAUTICAL ASTRONOMY. part of tlie sliip, since it lias been found that distiirl)ing the positions Las altered their rates.* When a chronometer is transported from one place to another, it should be compared, before and after moving, with another chrono- meter or a good watch, in order to ascertain whether its regularity has been disturbed. 627. A chronometer sliould be wound up at regular intervals, in order that the same parts of the machine may undergo the same constant action ; it sliould, therefore, be wound up at the same hour e\ery day. In winding, the key should be turned steadily, and about half a turn taken each time, and the watch should be wound close up. After winding, tlie chronometer should be examined, to ascertain that it has not stopped. In winding up a watch, the key alone should be moved, as to turn the watch itself is to increase the velocity of winding. AVhen a chronometer is wound up after running down, it is set a-going by giving it a small horizontal circular motion. When a chronometer stops, it generally alters its rate. 528. It seems generally admitted that the jirincipal cause of the variation of the rates of chronometers is change of temperature,f and accordingly, in some ships, the temperature of the chronometer- room has been regulated by lamps. When the ship changes her climate, the rates do not ciiange at the same time with the temperature, but some time afterwards. J 529. It has been found that magnetism affects the rates of chro- nometers (see a pacer by Mr. Fisher. Nautical Magazine, 18;37). Hence it follows, that the magnetism of an iron vessel may pi'oduoe similar effects. Their rates will certainly be affected by the proximity of apparatus generating or conveying electric cun-ents. 630. Chronometers are generally found to perform best at the * This depei\ds, however, chiefly on the position of the arm of the balance. t Captain R. Owen, while employed in surveying in the West Indies, found a fall of 14° in Fahrenheit's thermometer (from 82^ to 68^) accelerated the rates 1"5 a-day, and a fall of 20' (from 82° to 62^) accelerated them two seconds a-day. X Admiral Fitzroy, who employed in his surveys of South America the unusual number of twenty-two chronometers, observes, that the ordinary motions to which chronometers arc subjected, both from the incessant action of tlie sea and in transferring them from one vessel to another, scarcely affect the rates of good watches ; and that, in general, temperature is the only cause of the alteration of rate. (Journal of the Royal Geographical Society, vol. vi.) Sir E. Belcher, however, when engaged in the survey of the west coasts of North Ame- rica, found the chronometers of H.M.S. Sulphur very materially deranged by the jerking produced by a looseness about the rudder-head and from towing the Starling, her tender ; and observes, that when these causes were removed the watches performed admirably. In the Instruction R^glementaire pour les Batiments de la Marine Royale, &c. (An- rales Maritimes, 18-10), it is recommended that the chronometers should be held in the hand during the firing of guns, and that in transporting a watch from one place to another it should be carried in both hands, in order to avoid giving it suddenly a circular motion, which may be communicated by taking it up by a handle, or becket, at the top of the case. M. Givry considers that the rates of the chronometers of La Coquille frigate, commanded by M. Duperrey on a scientific expedition, were altered by the severe thunder-storms expe- rienced on the coast of Timor, in August 1823. — Memoire sur I'Emploi des Chronometres a la Mer, par A. P. Givry, extracted from the Annales Maritimes, Paris, 1840. It has been surmised that the hot and moist climate of the coast of Africa has sj^oedily disturbed the rates of chronometers; but Adm. Vidal and Sir E iJelclier, in several years' ej:psriencu, have recoguioed no such efTecl. TAKINO OBftllfVATIONB, 193 beginning of a vo) age ;* many subsequently become useless froni iiregulciriiy, and some fail altogether. They are liable, also, to change tlieir rates suddenly, and then to reassume the former rates in a i'ew dajs.f 531. Since there seems no reason why any cause which alters the rate of one chronometer should not alter tiie rate of another i:i tiie same manner, the agi'eeuient of any number of chronometers, however great, cannot be unreservedly admitted as evidence for the truth of the time which they shew. Their irregularities, however, in this res])cct contribute to the security of navigation ; for since one chronometer often gains while another, under exactly the same cir- cumstances, loses, the discrepancies prevent the danger of trustmg too confidently to any single result. CHAPTER III. Taking Obseuvatioxs. I. Observing Altitudes. II. OnsEUVATiONS with and withodt Assistants. III. Employment of the Hack Watch. I ^^ Finding the Stars. 532. In treating of observations with reflecting instruments we shall refer chiefly to altitudes, as most convenient for the purposes of illustration. If, however, for the horizon, we substitute a celestial body or any other point, what is said of altitudes will ap])ly, with certain obvious exceptions, to angular distance generally. The; details pro]K>r to the particula" obeervations will be found under their respenive beads. I. Obseuvixg Altitudes. .0.33. The observer will do well to accustom himself to obtani a single sight with accuracy, and not to depend upon the accidental compensation of errors due to want of care. It sometimes happens that a single sight oidy can be obtained, and no good estimate of its * Advantage was taken of this circumstance in tlie late survey of part of the west coast of Afric-i by Admiral V'iil.il. who, by direction of the Uydrographei, proceeded at once to run down the coast from Sierra Leone to Corisco Bay, and returned to Sierra Leone as c|uickly *.« piissilile. The whole Diff. Long, between tliese points, as nie««ured in both runs, agreed Kitliin 1>. t Captain R. Owen remarks, that most of his chronometers took tlius a jump of one oi tvro seconds in the daily rate, more than once during his surveys in the West Indies. Otuei sificers have made aimilar reiuuka. 19'i NAUTICAL ASTRONOMT. value can obviously be tbniied if the observer knows his observHliom by thoir j^eiieral result only. 1. At Sea. [1.] Above the Sea Horizon. 534. The instrument must be vibrated or swung, so that th« linage may skim the horizon, for the altitude must be measured to the jioint vertically under the body,* No. 487. 635. When the altitude is al)Ove 60", it may be observed both from the opposite point of the horizon and from that under it, l)y the common sextant. Half the difterence of the two readings is tiie apparent zen. dist., No. 432. By this means the dip, with the un- certainty to which it is liable, and the index error, are removed. As the a])parent dip is always uncertain, and as the rules given in No. 2U8, though generally true, do not always hold good for small differences of temperature, it will be advisable, whenever precision is required, to attend to this consideration. 536. It is, in general, taken for granted that the dip is in the same state all round the horizon. This supposition M. Arago, in discussing the observations made l)y Sir E. Parry in his first polar voyage, by Capt. B. Hall in the ('liiiui Sea. and by M. Ganttier in the Mediterranean and Black Seas, thinks there is no reason to doubt. ("Coim. des Terns," 1827.) Capt. Pitzroy found however a difference of 16' on one occasion ; and Capt. Bayfield informs me that he has often observed the dip not to be the same all round the hoi-izon, more jiarticnlarjy on the coast of Labrador and in the Straits of Belleisle, where currents of unequal temperature prevail. See also note *, p. ]9(i. AVhen circumstances allow, alts, should accoi-dingly be observed at opposite points of the horizon. The mean of two alts, in such cases may not, indeed, be exactly true, but it is probably nearer the truth than one of them alone might be. For the same reason it is advisable to select stars on opposite bearings. When both the alt. and its supplement are tlius measured, and the alt. is in a state of change (as will always be the ease exce])t when the object is on the meridian), the time must be noted at each of the two contacts; and the half difference of the alt. and its suppi is the apparent zenith distance of the centre corresponding to the mean of the times. When the altitude is below 60° a sextant of additional powers, or a circle, is in general necessary for this observation. (See No. 504.) 537. When the altitude of a body is near 90°, it is pro])er, before attempting to bring down the reflected image, to ascertain, by re- * WTien the 4th Adjustment. No. 495 (3), is not perfect, we look at a point of the honzon not directly under the sun. Hence a tuhe should he used to insure the eye and the oontact of the images being at equal distances from the plane of the instrument. On tha lame ground. Dr. Maskelyne recommends the ohsen'er, when without a tube, to turn oa *iu kwei while causing the image to skim the horizon. (Nautical Almanac, 1774.) TAKING OBSF.IlVATlONd. 19j forence to tlic zonitli, or the compass, tiie precise pmiit over \vl;L~h the body is vertical. 538. When fog- obscures tlie sea-horizon from tlie deck, a new horizon may often l)e obtained by descending the ship s side, or from a boat. See No. 550, note. 539. When the limbs of the sun or moon are indistinct, altitudes of the centre are obtained by bisecting the hazy or cloudy disc upon the horizon.* 5-l0. In observing tlie moon's altitude there is a choice of the upper or lower limb when she is at the full, and also when the line of cusps, or horns, is vertical. At other times her illuminated limb, whether it be the upper or lower one, must be brought dovvr. to the horizon. Mistakes may arise in observing the moon's altitude at sea by Tiight. When the sky under the moon is unclouded, the upper edge of the illuminated part of the sea is the horizon ; but at other times long dark shadows are projected on the water, which render it dit- tieult, and sometimes impossible, to diseein the liorizon. When the moon's alt. and its supplement are both measured, if she is full, or if the line of cus])s is vertical, her alt. may be observed as directed in No. 535. But in other cases the same limb must be referred to the point of the horizon under her and to that opposite; half the difference is then the app. zen. dist. of the limb observed, and the semidiameter must be applied accordingly. When the horizon under the moon is unfavourable for observa- tion, and the supplement of the alt. alone is employed, correct the angle observed for index-error and dip, take the suppl. of the result to 180", and apply the semidiameter as to the alt. taken directly. 541. The obscurity of the sea-horizon in a dark night renders it difficult to observe the altitudes of stars or planets; but in the twi- light, when the sky is clear, the boundary of the sea exhibits a strong dark edge, most favourable for observation. The difficulty of reading oflF at night is easily overcome by having a well-trimmed dark lauthorn, and a handy assistant.t Wlieii the alt. ot a slar or a planet is measured bolli from the horizon under it and opjiosite to it, half the diff. of the two angles is the app. zen. dist. If the supplementary arc alone is employed, correct it for index-error and dip ; the supplement of the result is the apparent altitude. 54'2. When a telescope is used the unemployed eye must be closed, but when the |)lain tube is used it should, when convenient, be kept open, because the image being seen by both eyes under the tame magnitude, one assists the other. This should be practised in observing stars at niglit. La Caillc recommends keeping the eye some minutes in complete * Mr. Fisher tclla mo tliat he hiv3 rejieateiJIy emplojej, with i-oiiip'oto success, ftltitudea of the sun faintly seen through watery cUiuds, wlien those who had been used to depend solely upon the [jerfectly defined disc hail despaired o( an ohscrvation altogether. In such cases the altitudes have not greatly dilTerefl from each other, and the mean of several has been quite eoual tj an ordniary observation of the limb. t Aimall electric light (halt candle power) io found useful. 196 NAUTICAL ASTRONOMY. darkness before observing stars at nigbt. (Guejirattc, " Probleiiies d'Astron. Naut." &c., torn. i. j). 20, 18:^9.) 643. Different ))owers suit different eyes. Too low a ])o\M'r does not magnify enongb ; too bi^b a one makes it difficnit to k<'t|i tlie object in tbe field on tbe least motion of tlie instrument. 'J'lie observer, tberefore, will employ tbose powers only in wbicb tbe advantage gained by a larger image exceeds tbe disadvantage of •ncreased unsteadiness. A plain tube, bowever, sbould be used in all otber cases, botli for directing tbe siglit to tbe prjper point of observation, and for detience against disturbing ligbts. 544. All observed angles are vitiated by tbe errors of tbe instru- ment enumerated in tbe last Cbtipter, Nos. 495, 4S)8, and 499. Again, eacb observer lias in general some peculiarity in tlic manner of observing, or in tbe quality of the eye itself, wliicli gives rise to a personal error, the correction for wliicb is called tbe ■personal eijuation. No. 175. 545. Besides tbese errors, altitudes taken at sea are subject also to otbers wbicb change with circumstances. 1st. Tbe running of tbe waves causes tbe horizon to be in con- tinual motion; 2d. The rise and fall of t!ie observer, both fnun tbe tilting of the vessel by the waves, and by her rolling, cause the dip to be in continual cliange. The effects of these alternating motions will, in taking two or three altitudes, in part disappear. .3d. The place of the visible horizon changes with the tempera- ture of the sea and the air. See No. 208.* Also, since the .sea- horizon is formed by the eminences of tbe waves, it sbould be higher in bad weather. -j- Besides tbese distinct causes of error, the motion of the ship disturbs the attention and efforts of the observer. 546. Tbe height of tbe eye should be ascertained with some precision, that is, within two or three feet, because an error in the dip causes an error of the same amount in tbe altitude. This is of most importance when the observer is very near tbe water, as tbe dip then changes most rapidly; thus, it appears in Table 30, that a change of three feet in the height produces, near the beginning tit the table, a change of more than 1' in the dip, but near the end only * M. Givry observes (" Memoire sur I'EmpIoi des Chronometres," p. 23), tljat wlu-u the sea is shoal near the horizon, the relation of the temperatures of the sea and the air hiiiig different from that at places where the water is deeper, may produce extraordinary rcfracticin: uil he attributes to this cause errors amounting to 8' in the time deducx-d from some altiluiles taken near the mouth of tlie Jeba, in 1818, although circumstances appeared at the time in every respect favourable for obaervation. M. Givry remarks, further, that extraordinary refraction sometimes takes place in the neighbourhood of sandy plains, the heated air of which, passing over tlie sea, produces partial inequalities of temperature ; and he adds, that small undulations in the horizon are always indicative of irregular refraction. t It is stated, "Voyage autour du Monde," 1840, by M. Du Petit Tliouars, in tha Venus French frigate, that the observations sliewed tliis. It is probable, however, that tha errors of observation due to the motica woiiW, in general, far cxjced tliat due to the above ■lAKINM OBSERVATIONS. 107 4". All altitude oliscrveil at the toj) of a heavy sea will ('iffer coiisi- Hnrably from anollier taken at or helow tlie mean level.* If the altitude be observed above the deck, as in the top for instance, tiie horizon will apjjear better defined, and the variations of tlic dip by the shij)'s motion will be less sensible ; also the dif- ference of temperature of the sea and tlie air ajipears to alfect tiie plaee of the visible horizon less as the observer is more (devated. Hence it would appear that altitudes should be taken fron\ aloft when convenient. 547. Some observations on tlie lieights, distances, arid velocities of waves have been put on record of late years. Sir G. (Jrey,i- in his voyage home from Australia in 1837-8, obtaitied niinierous measures of the distance and velocity of waves, amongst «Iiicli are ihe following : — Vel. .45 Na iS-7 HI ft. Vcl. 195 r-6 Lieut. AVilkes (" U.S. Exploring Expedition") found the liighest waves in a heavy sea off Madeira from 14 to "25 feet high, and theii velocity 23 miles an lioiir; and at another time and place, wiih a remarkably higii and regular sea, 32 feet, with a velocity of 26 miles. The highest waves observed by Sir Jas. C. Ross, in the Nortk Atlantic, were 36 feet liigii. The highest sea seen by M. Lazarev, in the Russian Expedition of Admiral Bellingsliausen, 1819, was in 56° S. and 103° E.. but he does not state the height. In the Naiit. Mag. 1848, p. 228, are the following observations taken near the Cape of Good Hope: — Heiglit 17 f. Dist. 35 fatli. Vel. 21 miles. 20 43 t" 5° 24 21 55 to 57 26 to 27 .W8. NN'lien the spectator nears or recedes from tlic celestial btxiy, by the progress of the ship, the eti'ect ])roduccd on the allitude IS the same as tliat of a motion in the body itself, since exactly the same appearances result from the motion of either while the other remains fixed. Accordingly, in all observations, in which, from the sensible change of altitude, the time requires to be noted at each sight, the progress of the ship is included in the observed change of altitude; and the place to which the observation corresponds is that at which the ship was at the mean of the times. * The liii(;l)t of waves is asccTtained by placing one's self at »uih a height on the vessel, or her rigping, that the tops of the highest waves which pass near the ship may be seen on with the liisiant wclldetimd horizon, at the instant when the ship is at the bottom of the hollow between two heavy seas. The height of a wave thus observed, that is, the differenro of level lietnifn the summit and the bottom of the hollow (which difference is twice the height of the summit above the mean leeel), is very nearly the height of the eye alwve tJie bottom of the same hollow, the ship at the instant of observation being upright. Tlie die* tance is measured, when before the wind, by a line with marks on it. t Goiernor of New Zealand. 1 am indebted to the author for these observatioiu, of which 1 hnd a few only reduced for the course aud rate of sailing ol the ship. 198 NAUTICAL ASTRONOMY. [2.J Altitudes above the Shore Horizon. 549. It often happens that the horizon is concealed hy the inter- fention of land, while the level surface of the water marks on the Bhore a distinct horizontal line, which is a substitute for the sea- liorizon, and is called a shore- horizon. When the distance of the shore-horizon is known, enter Table 36 with this distance and the height of the eye, and use the correction therein instead of the dip in Table 30. Ex. From the height 20 feet, observed 1 Alt. 2818' ) merid. alt. 28° iS', above a shore-horizon, Corr. — 7 » miles and a quarter distant. | Alt. corrected for dip 7s~T7 550. When the distance of the shore-horizon, or water-line, is not correctly known, it may be found by means of two altitudes, the one being observed from tiie deck, and the other as high as possible, at the same time. Divide the difference of the heights in feet by the number of minutes in the diff. of alts. ; the quotient is the number of feet sub- tending an angle of 1' at that distance. Look in Table 9 for tliid number of feet, and the cori-esponding distance is the distance required. Ex. An observer, at the height of 91 feet above the sea, observed the sun's alt. 41° 3/ »bove the water-line of the sea ; another observer, at the height of 22 feet, observed it 41' 25': find the distance of the water-line, and correct the alt. for dip. The diff. of the heiglits, 69 feet, divided by 12 (the minutes in the diff. of alts.), gives 5'7 feet, which answers, in Table 9, to 3 miles, the DiST. required. Then the cor. in Table 35 to 3 miles, and height 22 feet, is 5', which subtracted from the alt. taken at 22 feet, gives 41° 20', the Alt. corrected for Dip. But as this result, like the preceding, becomes uncertain when the distance is very small, it is always advisable in such cases to en- deavour to find, by descending, a natural horizon.* 2. Observing Altitudes on Shore. 551. Altitudes are well observed above the sea-horizon from a hill or cliff of known height. Nos. 544, &c. ap])ly, with certain obvious exceptions, to altitudes of this kind taken on shore. 552. In taking the altitude of the lower limb in the quiek.=ilver, the loxoer limb of the object is made to touch the upper limb of the image in the quicksilver, as reflection inverts the object. In taking the altitude of the upper limb, the image of the body is in like manner brought below the quicksilver image altogether. Hence, when the sun is rising, and the loioer limb is observed, the images tre continually separating; but when the upper limb is observed, they are continually overlapping ; and the contrary when the sun is falling. It is useful to attend to this, as it is sometnnes doubtful, cs))ccially with the inverting telescope, which limb was observed. * This is the practice rccommcnilnl, nii his own rvpciiciici bj Dr. Scorcsby, '■ Vojng* to the Northern Whale Fishery, \^12, l-ondiMi," p. 141. TAKING OBSERVATIONS. 109 56;3 It is ailvisiilile, when circumstances permit, to move tli* iiitlex a little too niucii, whether forwards or baci<wards, and clamp- ing- it, to wait tlie instant of contact while the instrument is in a state (if repose, in preference to making the contact by moving the tangent (.crew up to the instant of observation, because the material always springs more or less. Again, moving the tangent screw diverts a portion of the attention which should be devoted to the contact alone. At sea this is rarely practicable in any observation on account of the motion of tlie ship. .054. The roof of the quicksilver should be reversed at each set of three or five altitudes, in order to remove the effects of errors in the glasses ; one face is accordingly marked A and the other B, and these letters marked against the altitudes. The roof should obviously be used only when it cannot be dispensed with. 555. A stand for the sextant or circle, on sliore, is a great con- venience, and allows a liigher power to be used ; practice is, however, necessary, in order to derive the full advantage from it. 556. The accuracy with which a set of altitudes has been ob- served may, in part, be infei'red from their agreement with each other. For since the change of altitude in small intervals of time is nearly proportional to the intervals (unless the object is near the meridian), any considerable irregularity must be a consequence of an error of observation. The comparison of the differences of altitude, with their respective intervals, may easily be made by means of the Traverse Table, as iu the following example : — Ex. Observed altitudes of Arcturus in the artificial horizon. Time lo'. 5-43. Diff.^ j^,j -jo^^, ^^, Diff. .0 8 .7 \ ]\ 78 H 30 t' ,0 w 29 3 ,^ 30 57 10 14 20 ' 76 33 40 ** In Table 2, 2" 34', or 154', as D. Lat., corresponds to 44 as Dep. at 16". On the same page 3"' 12', or 192", as D. Lat., corresponds to 55 as Dep., which is near enough. 2"' si*. or 171", as D. Lat., corresponds to Dep. 49, the Diff. 44' is therefore in error, and the M «lt. about 5' too great. 557. Several altitudes are taken in immediate succession, on the Bupposition that they are liable to errors of opposite kinds; f()r, in this case, if one altitude be observed a little too great, and another a little too small, the mean of the two will be nearer the truth than either of them separately; and thus, by increasing their number, the effects of irregularities of observation will be much diminished in tlie general result. 558. But if the portion of time during which the altitudes are taken be too long, an error of a new kind will arise from the nneqiial variation of the altitude itself, which never, strictly speak- ing, varies at the same rate at the beginning, middle, and end of an interval. If a scries of alts., at observed equal intervals of time, be cleared of errors, and the differenoes between thcni be taken in succes- 200 NAUTICAL ASTRONOMY. sion, tliese (lifferciiccs will generally afford, in like maimer, liiirer- eiice? among themselves, which are calletl second differences ; and \t the observations be ]>rolonged, third differences will appear, and so on. When the 2d diff. is insensible, \ the sum of 2 alts., or ^ the ?um of 3 alts., or \ the sum of 5 alts., corresponds exactly to the middle of the time occupied in the observation ; but when the 2d ditf. is considerable, the arithmetical mean is in error by a quantity which is as follows: — The half sum of two alts, at the beginning and end of the interval differ from the alt. proper to the middle instant of the intei'val by l of the 2d diff. proj)er to the whole interval. The third of tiie sunt of tlie three alts, at the beginning, middle, and end of the interval, differs from the same alt. by Jj of the whole 2d diff. ; and the fifth jf the sum of 5 alts, at four equal intervals, by j'^ of the 2d diff. Ex. Lat. 5 1° 30' N. Decl. 22° 20' N. Hour-Angles. Alts. Diff. 2d Diff. 1st. qI' i6"'g« 60° 40' 8' 2d. o 20 o 60 34 35 5 33 ,. .Q. 3d. o 24 o 60 27 52 **■■* III •lih. o 28 o 60 19 58 7 54 , ,j 5tli. o 32 o 60 10 52 9 " The mean 2.1 Diff. is i' 1 1" for 4^" ; hence, as the 2d Diff varies as the square of the Interval (that is, is 4 times greater when the interval is douhled, 9 times greater when it is ♦.rehled, and so on), the whole 2d Diff. for 16'" is 4 times 4, or 16 times i' n", which ia 18' ^6". Then the mean of the 1st and 5th Alts, is 60° 25' 30", which differs from the 3d Alt. by 2' 22", or i-Slh of iS' 56". The mean of tlie 1st, 3d, and 5th Alts, is 60° 26' 18", which differs from the 3d by 1' 35", or i-i2th of 18' 56". The mean of the 5 Alts, is 5o'= zb' /li", which differs from the 3d by i' 11", or i-i6th of 18' 56". The error cannot be materially diminished by further increasing the number of alts. The correction for this error cannot be given in a concise and convenient form.* But in practice tlie intervals are not exactly eipial ; and even if they should be, the errors of observation wifl uCten conceal the 2d diff. When, therefore, from circumstances, altitudes can be obtained only at considerable intervals, it is projier to deduce a separate result from each. The 2d diff. of alt. disappears in two cases : 1st, when the object IS E. or W. ; 2d, wlien its motion is vertical. 569. The effect of the elevation of the s]iectator ujion the altitude observed in the quicksilver, is insensible in practice, since, even in the case of the moon, an elevation of a mile does not i)ruiluco a change of I" in her horizontal parallax. * The change 0' altitude in a very small portion of time depends on the latitude, and OB the azimuth of the object (see No. 669) ; but the 2d Diff.. or variatiun of the change of idt., which becomes conspicuous in a longer interval, di'|)cMils, furllu-r, upim the altitude itself, Xo exhibit this correction, therefore, a table ni treble enliy would l)e reijuired. TAKING 0USF.KVAT10N8. 201 II. Obskuvations with and without Assistants. 6()0. \\heii the arc observed is in a state of continual change, the ([uantity uieasiired corresponils to a particular instant of time. Wlien, therefore, the complete ohservation consists of varioua elements whose measures are required at tiie same instant, either the observer must have assistance, or he must himself obtain the "cveral measures in succession, and these must be reduceil afterwards to the same instant by calculation. When two or more altitudes at sea are required at the same instant, assistants have been employed to observe them. The inipro- ))riety of this custom w ill, however, appear on considering the nature of the errors of altitude (No. 545); for it is obviously impossible for an observer to keep the motion of the index so exactly adjusted to the irregular and often violent motion of the ship, as to be able to seize the altitude at command. 561. The assistant is useful chiefly in noting the time. An ob- servation of a set of altitudes, with tiieir times, for example, is con- ducted as follows: — (1.) The observer sets the index to the estimated alt. (No. 492); about i of a minute before he expects to complete the contact, he cries, " Look out!" at the instant of contact, he cries, '• Stop !" on which the assistant writes down the second, the minute, and the hour. The observer then reads off the degi-ee, nnnute, and division of the seconds, as 10", 20", 30", &c., which the assistant writes down. Three, Hve, or more altitudes make, generally, a set of W lien the assistants have watches shewing seconds, each takes his altitudes at leisure, and the whole is reduced to the same instant by calculation. (2.) The times are then added together, and the sum divided by the number of alts. 'J'he alts, are then in like manner added toge- ther, and the sum divided by their number is, when the second difference is not consideralde (3\o. 558), the alt. corresponding to the luean of the times. When the number of alts, is odd, and the intervals are nearly equal, the means will not ditier much from the middle time and its corresponding altitude. 562. When two sets of observations are taken by different jxr- sons, nearly at the same tnne, they are reduced to the same instant thus: — The difference or change of altitude (or other angular measure) in the time occupied by the observation is given ; then the interval between the given mean of the times, and that !o wiiich it is j)ro- jioseil to roiluce the observation, being tbiirid, the quantity to be applied to the altitude is determined by j)roi)ortion. For accuracy, the chanu:e of alt. must be properly computed by No. 6()!) or 671. .06;). Tiio ..b^(•rv(•r shuuhl, liowcvr, take tlie whole observation himself, a'ld he will liicn learn to otimaie hi^ results ai their leal 202 NAUTICAL ASTRONOMY. value, of wliich lie can bo no judge when they are taken l)y other jiersons. Wiien tlie observer takes his own time, he holds his watch in his hand, or places it either where he can obtain sight of it readily, or where he can hear it ticlv plainly. In the latter case, the first beat alter the instant of contact he counts 1, tiie next 2, &c. ; then, looking at the watch, he counts on till the second hand arrives at a marked number of seconds, as 10, 15, (fee; he then writes down these seconds, and after them the number of beats counted, to be sub- tructed. If the observer can count 10 or 20 seconds without an error of more than 1* or 2% he may put the watch wherever it is most conve- nient to inspect the face, and thus avoid the principal difficulty in taking the entire observation himself, especially at night. He then reads otl" the alt., and sets it down. The sum of the beats is to be deducted before the mean of the times is taken. Most watches beat 5 times in 2% or each beat counts 0'-4. Ex. After the instant of contact, i.). beats a.-e counted ; the seconrt-liand is then ■t 50', the min. 41, and the hour 10, and so on, as follows ; — 0" 42" 1°' subtrae t 14 hnu. 44 32 46 111. E.MPLOY.MENT OF THE IIaCK WaTCH. 564. This is a portable chronometer, or good watch, used for observation, to save moving the standard chronometer. Since the watch and chronometer will not in general go exactly together, they must be compared both before and after observation, in order to find wjiat time the chronometer shewed when the observation was taken. Thus, Within 5 or 10 seconds of a whole minute by the watch the observer tells the assistant to " look out" on the chronometer. At the minute he cries "Stop!" wlien the assistant writes the times, and takes their differences. This should be repeated two or three times, and the mean result employed. The observer can compare alone, by counting the beats of the chronometer till the expiration of the minute. If the difference between the watch and the chronometer be the snme before and after observation, the time of observation by the chronometer is at once deducd from that by the watch ; if not, a correction must be apjilied, aa in the following exaiu|>)<?: — TAKING OBSKRVATIONS. 203 u>. AflerObs. IntCTV.iIn. Waloh ^i" : r" o» Chron. lo 31 184 Diff. 7 20 18-4 7 20 2 17 r3 1 line of observation by watch, 3'' 32"' 37*: required the time of do. by chron. The watch here has lost 3"3 on the chron. in 52". The observation taking place 21"' 37« by watch, alter the tirst comparison, we have 52'": 3" 3 :: 21'" 37*: i''"4, the loss of the watch on the chron. at the time of observ. ; this, added to 21'" 37", gives 21'" 3S'''4, which, add d to ici'31™ i!i«-4, gives 10" 52"' 56-S, the Time by Cubon. required. /jfio. M'lieii the times by watdi are separated by coiisiilerablo interval?, and the rate of the watcli is larjre, each time may requi.re to be thus cori'ected for its jirojicr gain or loss. IV. Finding the Stars. 566. Tiie most conspicuous stars have been designated, from remote antiquity, by names; besides which, tlie stars in each con- stellation or group aie distinguished, for reference, by letters and numbers. The letters chiefly used for tiiis purpose are the small letters of the Greek alphabet, which, with their names, are written as follows: — a alpha ^ zeta X lambda » pi f phi f> lieta n eta /i. mu {To x ^\ y gamma ( tbcta » nu <r sigma 4 P^i } delta I iota ? ksi t tau ..omega f epsilon x kappa e omicron v upsilon 507. In finding any star in the heavens, it is necessary to refer to some one star or constellation as known : the Great Bear, called filso by the Latin name Ursa Major, a constellation of the figure t-hewn below, in the northern part of the heavens, and consisting of Beven principal stars, is the most convenient for the purpose. i'ult St.ir. 204 NAUTICAL ASTUOXOMT. T]ie two stars o and /3 point nearly to the Pole Star (or rolaris)^ and are hence called the Pointers. This star will uot easily be mis- taken, as it appears always in the same place. A line from Polaris through t] (the last of the tail) passes, at 31° beyond t], through Akcturus, one of the brightest stars. A line drawn from Polaris perpendicular to the line of the Pointers, and on the opposite side to the Great Bear, passes, at 48" distance, through Capella, one of the brightest stars. In this same line, about the same distance on the opposite side of the pole, is a Lyrw, or the bright star in the Harp, called also Vega, and also by sea,nien Lyra, a large white star. At one-third of the distance from Arcturus to a Lyrce is ALPHACCA,the brightest of a semicircular group called the Northern Crown (Corona Borealis). A line drawn from B (the faintest of the Great Bear) Ijhrough Polaris, passes through the constellation of Cassiopeia. About 23° to the eastward of a Lyrce, and about the same dis- tance as this star from Polaris, is the bright star in the Swan (or a Cygni). Deneb. A line from Polaris passing between this last and a Lyrce, pro- duced to an equal distance beyond them, passes through Altair (a Aqtjila;), a bright star between two small ones, the three lying in the direction of a Lyrw. The line of the Pointers, carried through the pole to about 62° beyond it, passes through /S Pcgasi, called also Scheat, and about 13° further, through Marxab (a Pegasi). A line from Polaris, drawn between Capella and a star near it to the eastward, passes to the westward of the constellation Orion The two northern stars of the four at the corners are the shoulders, the northernmost of which is Betelguese, or a Oriimis. The brightest of the two southern stars, the feet, is called Ri(;el. Iu the middle are three small stars forming the belt, the northern- most of which is nearly on the equntor. About 25° to the northwestward of the belt, and not fiir out of the direction in which it points, are the Hynclps and Pleiades iu Taurus; in the former cluster lies the red star Aldebaran. A line from Aldebaran through the belt passes, at about 20° on the other side, through Sikius, the brightest t)f the stars. tSirius, the eastern sliouldiT, and Pjuicyon (to the northward of Sirius and eastward of Orinn), furiii an eqiiilaicrat triangle. Nearly midway between Uiion and the Great Bear are the Twins, Castor and Pollux (the southern and brightest), about 4;° apart. The line from Polaris to Procynn passes between thein. A line from Rigel through Procyon passes, at an equal distance bi.'vond, to the northward of Regulus. 8 and y Urs. Maj. serve as pointers for Regulus. A line dra "n from Procyon through Regulus, at nearly an equal distance beyond it, passes through /3 Lcouis, or Denebola. BUnnHDIXATK COMrUTATIONS. 205 Aline from 8 Urs.Maj. tlirough Ecgvlan, jiasscs, at r!0''bpyon(l, tliniugh Cor Hydr^. A liue iVoin I'vlaris through f Vrs. Maj. passes, at 70° distance, tlirough Si'iCA ViKGiNis. A liue from the last star in the tail of the Great Bear through Arcturus will lead to a and /3 Librcc. Arctarus, Spica, and Veiwhola form an equilateral triangle. A line from Regulus through Spica passes, at 45° distance, through Antares, a very bright and reddish star. A line from a Oriunis (Bett'lguem;) through Aldebaran passes, at Go" distance, through a Arietis, not a very distinct star. The Southern Cross is about as far from the South Pole as the Grent Bear is from the North Pole ; a is the foot, and 7 the head. To the left of the Cross when on the meridian and pointing towards it are a and /3 Ci'titauri, both of the first magnitude. A line from Scheat through Markab passes, at 45° from Markah, through FoMALiiAUT, a very bright star. Sclieut -dud a ANDROMED.f;, called also ^//</i era fe, form the north side of a square ; Markab and Algenib on the south side. AcHERNAR, Fomalhaut, and Canopus, are in a line, and nearly equidistant, being about 40° apart. 5(j8. When a few stars are known, the rest are easily found by the times of their Meridian Passages, Table 27, and their Declinations, Table 03, as described in No. 482 (8). A star may also occasionally be identified by means of its aUiluJe, or azimuth, computed rougldy. CHAPTER IV. SuROIiOINATE CoMl'lITATIONS. L The Greenwich Date. II. Reduction op the Elioments IN TUE Nautical Almanac. III. Conversion of Timks. IV. Hour-Angles. V. Times op certain Phenomena. VI. Altitudes. VII. Azimuths. 5G0. Such parts of computations as are common to more opera- tions than one are collected, both to avoid repetition and for facility of reference, in this chapter, which contains also some smallrr computations not relating directly to the principal divisions of the stibject.* • Ci-riiiin compiilali'ms in this cliaplcr, though not of iinmi'ciiate ajiplicution in Ui« pri-avnt volmiic, inuv lie tuuiul usilul lur tlio i'ur['oses of vci'itifnti'Hi. 206 NAUTICAL ASTRONOMT. I. The Greenwich Date.* 1. Conversion of Arc and Time, h/0. To turn Degrees and Minutes into Time. Bi/ Inspection.— {\.) To the wiiole second. Enter Table 68 oi 60 with the given arc, anil taiie out the liour, minute, and second. Table G8 shews the time to the nearest two seconds. (2.) To parts of seconds. Take out of Table 17 the hours, miiuites, seconds, and parts corresponding to the given degree, min., Hud sec. Ex. 1. Tuin 36° 1 1' into Time. In Table 68, or 69, 36" 11' is seen to t 24"' 44" in Time. Es. 2. Turn io!°4i'45" into Time. Ans. by Table 69, 6i'46"' 47" in Time. Ex.3. Tarn 134° 52' 9"-7 into Time. In Table 17, 130', S''4o'" o' S^' Time required 8 59 2S-65 571. J3i/ Computation. — Multiply the arc by 4; this turns the degrees into minutes of time, the minutes (') into seconds of time, and the seconds (") into thirds of time.f Ex. 36° 11' multiplied by 4, is 144'" 44*, or z"" 24'" 44' in Time. 572. To turn Time into Degrees, Minutes, and Seconds of .Arc. Bi/ Inspection. — (1.) To the nearest second or two seconds. Employ Table 68 or 69. (2.) To parts of seconds. Take out of Tabic 18 the deg., min., and .sec. corresponding to the hours, luins., and sees, of time. .073. By Computation. — Turn the hours into minutes, and divide bv 4 ; the quotient is the deg., min., and sec. Ex. 1. 2'' 24"' 44" are 144'" 44", which, divided by 4, gives 36° 1 1' in Ahc. Ex. 2. 5'' 20"' are 320'"', which divided by 4 gives 80° in Aac. 2. Deduction of the Greenwich Date. 574. The Civil Date begins at midnight. No. 480 ; the Astrono- mical Date begins at noon : thus the civil date Oct. 1st, 3 p.m., is the astronomical date Oct. 1st, 3''; but 11 a.m. on this day, civil date, is the astronomical date Sept. 30th, 23''. In most cases it is necessary to refer to the astronomical time at Greenwich, or the Greenwich Date, No. 481, because it is for the time at this meridian that the elements of astronomical calculations, which are in perpetual change, are given in the Nautical Almanac. The Greenwich Date is always mean time., unless the contrary be expressed. At sea, however, it is often convenient to deduce the Greenwich Date in App. Time. * The term Gremwich Date, used always by Dr. Inman, is preferable to Greenwich Time, because it is essential to note the day as well as the hour. t The reason of these rules will appear on considering that dividing 360° into 24'- gives 15" for 1 hour, 15' for 1"', and 15" for 1'; and further, that to multiply by 00, and at the tanin time to divide by 15, is th- same as to multiply by 4 ; and to multiply by Ui ami to divide by 60 is to divide Dj 4. STJBORniNATK COMPUTATIONS. JO7 575. To find tlie Greenwich Date by the Chronometer: — Since tiie clironometer is regulated to GnBenwieh moan time. apply the gain or loss up to the time proposed. No example is necessary, as this is no more than the common jirocess of" alLo\vin<; for the error of a watch. 576. To find tlie Greenwich Date without the Clironometer: — (\.) In W. Long-. Find the Astron. Date, No. 574 ; add to it the Long, converted into time. No. 570. If the sum amounts to or exceeds '24", deduct 24'' and reckon the time on the next day. Ex. 2. June 4tli, 5'' iS"' a.m., Iuiir. 130 W.: find the Grefiiwich Date. Astron. Date, June 3d, 17*' 18°' S»"W.: Astron. Date, June 3d, 3'' 30" Greenwich, Jun: 5 3+ Greenwich, June 4tli, 1 58 (2.) In E. Long. Find the Astronomical Date, No. .574; subtract from it tiie Long, in time: the remainder is the Greenwich Date. If the Long, be greater than the Astron. Date, add •24*' to this last, and reckon the time on the preceding day. Ex.3. April 15th, 4'' 17"' p.m., long. Ex.4. Dec. 31st, 6" 57"' .4.M., long. 18" E. 40" E. Astron. Date, 15th, 4'' 17"" Astron. Date, 30th, iS'' 57"' 28 , — I 5: 40 ', —2 40 Greenwich, April 15111, 2 25" Greenwich, Dec. 30th, 16 17 (3.) When it is noon at the ))lace. In W. Long, the Greenwich Date is the Long, in time. In E. Long, take the Long, in time from 24'' : the remainder is the Greenwich Date on the preceding day. Ex. 5. February 13th, noon, long. I Ex.6. March 31st, long. 91° E. 122' W. I Long. 61' 4° Grkenwich, Feb. 13th, SI'S'" P.M. | Greenwich, March 3cth, 17 56 577. It is easy to ])erceive, on all occasions, what the Greenwich Date must be, by proceeding from noon at the place. Thus, in Ex. 2, when it is noi>n in 130° \V it is 8" 40" lateral Greenwich- ht when it is 61' 42'" UJore n.ion at this place, it i , 6" 42"' belc re 81' 40"', or i'' slC'" Greenwich, on the same day. Ex. 4. When it is noon in long. 40° E., it is 2'' 40"' before noon at Greenwich ; hence, when it is e"" 57'" a.m., or 5" 3'" before noon at this place, it wants 2'' 40'" and 5I' 3'", or 7'' 43" of noon at Greer.-»ich on tliis day ; or it is 16" 17"' on the day before. II. Rediction of the Ei.ement.s in the Nautical Almanac. 578. This Reduction is effected by Inspection, or by Logarithm*. No. 507. When extreme precision is required, a furtiier corrtjctioii is u(»c«;<»ary, on account of 2d Differences, No. 598. 208 NAUTICAL ASTRONOMY. 1. Reduction hy Inspection. [1.] The Suns Declmatim. 579. At Se:i,.—{\.) At noon. Take out of the Nautical Almanac, p. I., or Table (iO, the sun's decl. at noon of the day, and note whether it is increasing or decreasing ; take out of Table 19 the correction for long., and apply it, as there directed, to the decl. at noon. If the correction, when subtractive, exceed the decl. at noon in the table, the difiference is the decl. of the contrary name. Ex. 1. Nov. 13th, 1902, long 64° W. : fiiul the decl at noon. .Sun's decl. 13111, noon, I7°4S' S. (incr.) 64° VV. Table 19 +3 Rtn. DEcr.. 17 51 S. Ex. 2. March 20th. 1902, long. 178° W.- find ihe Sun's deil. at noon. Deel. 20ih, noon o 25' .S. {<lccr.) 178° W. Table 19 _-I2 Rei.. Decl. o 13 S. Ex. .S. June 20ih, 1902. long. 120° \V.: find the decl. at noon. Heel. 201I1, noon, 23° 26' N. (iwcr.) 120° W. Table 19 o Red. Decl. 2^26 N. I'x. 4. Sept. 22d, 1902, long. 167° \V.: finil the Sun's decl. at noon. Decl. 22d, noon, 0° 35' N. (ilecr.) 167° W. Table 19 _- 1 1 REn. Decl. o~24 N. Ex. 5. Aug. 6th, 1902, long. 85° E. : find Sun's decl. at noon. Decl. 6th. noon, 16' 54' N. (,h-cr.) 85° E. Table 19 +4 REn. Dec L. 16^ 58 N. Ex. 6. W^rcb 20th, 1902, long. So" W. : find Sun's decl. at noon. D.cl. 20th, noon, 0° 25' S.{ilecr.) So' W. Table 19 _-5 Red. Decl. o 20 S. When the declination at noon at Greenwich is 0" 0' in east long , the correction is the decl. of the same name as that of the day befcrre; in west long, the correction is the decl. of the same name as that of the day after. (2.) At a given hour. Correct for long, as above, and then apply the correction for the hour. Ex. 1. March 21st, 1902, long. 123° \V. at 3'' P.M. : find the decl. Dec!. 2Ist, noon, 0° l' S. (Air.) 'V- :?[ J" Red Decl. o 10 N. For 3'' A...1. the corr, «ill he Cor 9'', or 9', suUiacth-e, and the Decl. :s 0° 2' S. Ex. 2. Feb. 1 2th, 1902, long. 78° E. a 7'' 50"' r.ji.: find the decl. Decl. I2th, noon, 13° 54' S. (<Zw.) 78' E. +4;, Urn Decl. 13 52 S. For r 5°'" '-^i- 'l'«-- <•■•■■■'■. is that fo 4' 10'", or 3', uiljitive, and ihe Dei l. i 14° I' S. 580. Accurately. — (1.) Find the Greenwich Date.* Take out of the Nautical Almanac, p. J I., the decl. for noon of the same and the next days, and take the diff. between them, or the Daily Variation. When the declination changes its name, the daily variation is the eum of the two declinations. (2.) With the Greenwich Date and daily variation take out the proportional part from Table 21. SUHORDIXATK roNriTTATtONS. L^OS.V The reduction of the elements in the Nautical Almanac can aha be eOected by using the Hourl}' Variation given on p. I. of each month : taking care always to use the elements for the noon or hour nearest to the Greenwich Date. Several examiiles given umier Nos. 579-584, and 5^32 are thus worked : — 1. Reduction by Hourly Variation in Nautical Alriituiac. 579, [1.] The Sun's Declination. Ex. 1. Nov. 13th, 1878, long. 64° W.: find the decl. at noon. Long. 64° \V. = 4'' i6"' = 4''-3 Hourly Var. 39"7 x4-3 = 171" or +3'. Sun's decl. 13th. noon, 18° l' S. {iiu-r.) 64° W. Table 19 +% Eed. Dkcl. i5> 4 .s. (2.) At a given hour. Ex. 1. Jtarch 21st. 187S, long. 123° \V., at 3'' P.M. : find tbe decl. Long. 123° W. = 8'' 12"" + 3''= Ii''i2"= ii'^. Hourly Var. 59"-i x 11-2 = 663" or +11'. Decl. 2lst, noon, 0° iS' N.fiiicr.) I23nv. +8'1 3" +3' J _ Red. Dkcl. o 29 For 3' A.M. the corr. will bo for 9'', or 9', suhtractive, and the Decl. is o' 17' N. Ex. 4. Sep . 22d, I S-S, long. 67° W.. u the Sun- decl. a' noon. .ong. 167° W. = ii" 8" =11'- 1 lourlvVar. ss-'-sx 11-1=6.19" or - 11'. Decl. 22d o'^ 16' N .{de cr.) 167° W. Table 19 -II Ked. Decl. IN 580. (3.) Ex. 1. May gth, 1S78, at 11'' 30'" mean time at Greenwich : find the Sun's declin. Hourly Var. 39"-s x ii'^Cor ii"-s = 454"-3 = +7' 34"'3 Decl. 9th, at noon, 17 2 3 58 -9 N. Ked. Decl. 17 3! 33 2 N. Ex. 2. Feb. 12th, 187S, long. 78° E., at Long. 78°E. = 5''I2'"- 7'-5o" = 2''38'" = 2'-6. Hourly Var. 50"-! x 26^ 130" or - 2'. Decl. I2th, noon. 13° 38' (S. (dar.) 78° E. +4' I 7" SO" -6'/ .^ Hed. Deci.. 13 36 S. l-'or 7'" 50"' A.M. the corr. is that for 4'' lo". or 3', additive, and the Decl. is I3°45'S. 1S7S, IS* 27™ mean 582. E.^. 2. March 21 time at Greenwich : iS"27'"-24i' = 8"33"'or8'-55. Hourly Var. 22d, 59"-2 x 8''-55 = 506 '-2 = - 8'26"-2 Decl. 22d, at noon, 041 42 6 N. Red. Decl. o 33 16 -4 N. [2.] TJte Suns Right Ascension, Ex. 1. June 6th. 1878. at 8" II" a.m mean time, long 1 7° W. : required the ?un K.A. Astron. Time, June, t,'' 20'' li" Long. 17° W. + I 8 Green. Time, June, 5 21 19 21"' i9"'_24'' = 2'' 41"" or 2'-7. Hourly Var. 6th, 10-31 x 2''-7 = - 27-8 R.A, 6tli, 4 57 26 4 Reo. R.A. 4"56~58 6 583. [3.] The Equal. Ex. 2. Nov. 29th, 1878, long. 103° E. tt apparent noon : find the Equation of Time. Astron. Time, Nov. 29'' o' o» 103° E. -6 5 2 Green. App. T.. Nov. 28 17 8 1 7'- 8'- - 24" = 6" 52»or6''-9. Hourly Var. 29th, o" 89 x 6'"9 = + 6'l Equation 29th, at noon, — 1 1" 30 o Kp.d. E<j. of T. -ni~36 "i Ex. 2. March 22d, 1878, at 2'' 20" p.m. mean time, long. 43° £.: required the Sun' R.A. Astron. Time, March, 22* 2' 20" Long. 43° E. -2 52 Green. Time, March, 21 23 28 23" 28"-24'' = o" 32" oro»-53. Hourly Var. 22d, 9* 09 x o' 53 = —4' 8 R.A. 22d, o 6 24 7 Red. R.A. o 6 19 9 inn of Time. Ex. 3. Dec. 25th, 1S78, lon<T. 18" W. a S" O" AM. (app. time): fiud ihe Equali.,1 of Time. A.stron. Time, Dec. 24'' 17'' o" 18° W. +1 13 G.een. Time, Dee. 24 18 12 18'- i2«-24'> = 5''48'"or ■i'-S Hourly Var. 25th, i* 25 x 5''-8= -7'-3 Equation 25th, at noon, +0" 20 -o Red. Ei, of T. +0 12 7 [Tefaceiy. 203. SUBORDIXATE COMPUTATIOXS. 209 (.".) \Vl>on the first deel. is increasing, ndd this prop, part to the (led. at noon ; when decreasinc) , subtract it. If the prop, part, when subtradive, exceed the dech itself, the difference is the decl. of the contrary name. ch 21<it, 187S. 15'' 27'" ine:m Ex. I. Mny gtli, 1S78. at 11'' 30™ mean iiiie at Greenwich ; find the Sun'^i decliii. 91I., Tagell., N.A. 17° 23' 58"-9 N. loth, I7_39 47 •4N. Daily Var. 15 48 ■$ II'' 30"", var. 15' 30" 7 25 -6 185 8 -9 + 7~'34 -5 Qth, at noon, 17 23 58 -9 N. Keu. Decl. 17 31 33 -4 N. Ex. 2. ila lime at Grienwicli : find the .Su 2Ist, Page II., N.A. 0° 18' 2"-4 N. 22d, o 41 42 -6 N. Daily Var. 23 40-2 15'' O"', var. 2j' 30" 14 41 -2 10 -2 6 -4 27'", 23 40 26 -6 ^+15 l4"-2' 2 1st, at noon, O 18 2 -4 N. Utn. Decl. o 33 16 -6 N. The sun's decl. changes nearly 1' an hour, or 1" in l", in jNIarch and Sept. ; hence, to ensure it to 1" in the extreme case, the Green- wich Date must be true to 1™. The id. ditf. (see No. 5'J8) is 20" a-day in .Tune and December. The greatest error of omitting it is then ^ of 2(1", or 3". [2.] The Sun's Right Auenmn. 581. Appro'ximo.teif/. — Find it in the Nautical Almanac, or from Sidereal Time in Table Gl, for noon. See Note to p. 21 1 and p. 421. Ex. 1901, April 2Ist, find the Snn's Right A'cnsicn. Sidereal Time, April 2l.st, ,1. 55'"4 -i'"^ Equation of Time=l'' 542, Sun's II. A. 582. Accuratdji — (1.) Find the Greenwich Date. Take out of the Nautical Almanac, p. II., the E.A. for noon of the same day and the ne.\t. Take the difference between them, which is the Daily \'ariatinn. When the first K.A. has 1!3'' and the second O", add 24'' to the second, and subtract the first from it : the remainder is the Daily Variation. (2.) With the Greenwich Date and the Daily Variation find the proportional part from Talile 21. (3.) Add this prop, pari to the fir.st K.A. : if the sum exceed 24'', reject 24''. June 6(h. 187S. ,lnng. I7''\V.-. re^ Long. 17° W. Green. Time, June, R.A. 5th. Page II., N.A,, Ex. 2. March 22d, 1S78, at 2'' 20"' mean time, long. 43° E. : required the : ll.A. Astron. Time, March, 22'' 2'' 20"' Long. 43° E. ^2 52 Green. 'I'ime, March, 21 23 28 R.A. 2Ist, Page II., N.A., 22d, Daily Yar. 23'' o'", var. 3"' 30' 28". 3 3S o" 2'" 4 6' -4 o 6 24 7 '_3' 3S-3 3 21 -2 S-o __4 -2 + 3 3.i -4 o 2 46-4 o 6 ig 8 210 NAUTICAL ASTRONOMY. When the R.A. in the tables is 0, the prop, part is K.A. required. The greatest daily change of E.A. is 4™ 30' in December; the smallest, 3" 30' in September. [3 ] The Equation of Time. .583. At Sea.— (I.) Find the Greenwich Date. Take out the equation of time from the Nautical Almanac, p. I., or Table 62, for the same day and the next. When both the equations are directed to be added, or both to be subtracted, take their difference : if one is to be added and the other subtracted, take the sum : the result is the Daily Variation. (2 ) With the Greenwich Date and the Daily Variation find the correction or proportional part by Table 21. (3.) When the first Equation is increasing, add the prop, part ; y?hen decreasing, subtract the lesser from the greater. If the prop part, when subtractive, exceed the first Equation, their diff. is the Reduced Equation, and is additive or subtractive according to the direction for the second Equation. at 3'' Til Astror 41° W. 25th, 1902, lonp. 41° W. !ipp. tinje) ; find the Equa . Time . T. 25th, •,Jui,e, , June, 25" 3" + 2 Daily Var. 6" 12'", ^a^. 13- 25th, noon, Heu. E«. of 1 Ex. 2. N 1-2'" ir-0 f 2 24 'O 13 -o 2 14-0 29th, 1902, lonjT. 103° E. Dn : find the Equation u{ Astron. Time, Nov. 103° E. Gricn. Apji. T. Ncv. Eq. T. 28th, 29th, Daily Var. 17>'8"'. var. 21' 2Sih, noon. Uld. E«. Of T. 29'' 0° C" -6 52 28 >7 8 II -50 •0 11 29 ■0 21 •0 - 15 •I — II .so •0 -II 34 ■9 As the Equation of Time is generally required for a particular hour, the above method by Table 21 is more convenient that that by Table 20, in which the correction is given corresponding to the longi- tude, and the time at ship, without reference to the time at Green- wich. The first example worked by Table 20 will stand thus (uo STTH0I!1>INATK 211 further explanation being necessary, as (lie table is entered precisely like Table 19):— hx. 1. June 25th, 187S, long. 41° W. »t 3" 28" P.M. Ex. 2. March 26lh, 1S78, Ion?. lo.j" V), at 7" 42" A.M. (app. time). Eq. T. 25th,p.I,N.A. +2'"lS-5 ^ 26tl., +2_3I_^ Daily Vur. 127 41° W. +l-4\ ,~ 3" 28... +.-8) +3« Eq.25.h. +2 18-5 Kki.. Eq. OF T. +2 217 Astron. Time, March 25'' 19'' 42'" Eq, 25th, +6" 4-2 26th. +5 45 7 Daily Var. 185 12" 0" -9" 2) 7 4^ -5 yf- -9-5 109'^ K. + 5 b) 25th, noon. +6 4-2 i?K„. E«.ofT. +5547 .584. Accurately.— Vroceed as directed in No. 583, with more attention to precision in the several quantities. Ex. 1. Gr<-en. Date, June 25111, 1S7S, 5" 1 1- (app. time) ; find the Equation of lime. Ex. 2. Green. Date, Deo. 24th 1878. 15" 49" (app. time); find the Equation of Eq. 25th, page I., N.A. 2"' 1S-5 26tli, 2 31 -2 Eq. 24ih. N.A. -o'" io"o 25111. +0 20 -o Daily Var. 12 7 Daily Var. _30'3 6"o-,var. 12-7 3-2 1 1'", do. -l + 33 Eq. 25111. +2 18-5 15" 30™, var, 30- 19-4 19-, do. -4 -19 8 24th, Eq. -0 10 -o Ked. Eq ofT. +2 21 8 Kkd. Eq. ..fT. -o 98 [■>.] The Sultreal Time.* .58.1. Take from Table 2." the Acceleration corresponding to the hours, minutes, and seconds of the Greenwich Date; add them to the Sidereal Time at the preceding mean noon, from N A or Table 61. When the sum exceeds 24*', reject 24''. Ex I. Green. Date, Ni.v. IM. 190I, ^ Ex. '2. Green. Date, \[ari(i 23rd, 1901, jh 41"- 39>. find the .Sid. Time. By I 2o" 36" 57": find the Sid. lime. K/ 'J able* 61 and 23. I N.A. Sid. T. mean noon, Nov. 1st, 14** 40'"*3 I Sid. T mean noon, March 23d, o** I™ 7' o Accil. 3'' -5 j 20" 3 17 1 41- •. 36- 5 -9 3h' -o 57' -2 Ri;d. Sin. Time 14 40 -9 ' Rm Siu. Ti.mb 04 30 2 [5.] The Mnna's Horizontal Paallai. 586. At Sea. — As the Moon's Horizontal Parallax does not change more than 27" in 12 hours, it may be, in most cases, taken out of the Jsatitical Almanac at sight. .587. Accurateh/ ~{l.) Find the Greenwich Date. When the (freenwich time is less than 12", take out the hor. par. for the noon and miflnight of the given day; when it exceeds 12 ', take out the quantities for the midnight of the same day and the noon of the next. Take the difference between them, which is the variation in 12 hours. • The Sun's RiL'ht Asccn.sinn mav bo found mnghlv thus: -To the Sidereal Time in Table 111 apply the Kq of Time from Table BJ, a..i there rtirccted : tor ex., the Sidireil Time ou Ni>v. 1st, iQor. is 14*' 40""3. tKe l'!q. of Time is 16'" 3 .^iih. ; hen'-e, suhtractin^ i6'"*3 tVum 14I1 4o'"-3. yiies 14'' 24"', the Sun's R A. lequireJ. 212 NAUTICAL ASTRONOMY. (2.) Enter Table 21 with the Greenwich Time and the 12-hourly var., and take out the proportional part. When the horizontal paral- lax is increasing, add this prop, part ; when decreasing, subtract it from the horizontal parallax at the preceding noon or midnight. Ex 1. Green. Date, Jan. isth, 1S7S, Ex. 2. Green. Date, Aug. I2th, 1878, ,'.ll-: required the Hor. Pnr. 15" 28'" : required the H or. Par. H.P. I5ih, noon, 57' 45"-2 H.P. 1 2th, midn. 54' 56"-9 15th, midn. 58 12 7 I3fh, noon. 54 45 9 V.r. in ,2^ 27 -S Var. in 12\ It 5'"'". var. 27"-S +11 -9 3^2S", vnr i."-o -32 IS'li. noon, 57 45 -2 I2lh, midn. ?4 56 9 Kko. ]Iou. Pak. 57 57 , Rkd. Hon. Par. 54 53 7 When necessary to correct for latitude (No. 437), see Table 41, 892. [9] Ex. :!. Green Dale, Sept. nth I" 47"' : find Venus' K..\. K..'\. Sept. nth, nooci ii''37' Hourly Var.* nth, 5'-ixii'"-8 I Red. R.A. Venus II 36 Kiiiht As h, 1903, .'/ VeH'j^. Dat( 22" 47-°: find Venus' K,.\. R,.A. May 5th, noon Hourly Var.'* 5tli, I2*-9 X 22-8 Red. R.A. Venus 5" 17" 1903. 49-3 54-1 43-4 593. [10.] Declinati.m of Vti Ex. I. Green. Date, Sept. iitli, 1903, [l' 47'» : find Venus' Declination. Hourly Var.'* Sept. i iih 23"-9x III- 47'"= -4'42"-o Decl. Venus, Sept. nth, 6° 51 30 -o Red. DtrL. Venus 6 46 48 -o Ex. 2. Green. Date, Sept 11'' 47'°: find Venus' Declina Decl. Sept. Ilih, „ Sept. 1 2th, Daily Var. 11" 30'", var. 930 6-3 17-, 9 36-3 6° 51' 6 41 ?_ s, Sept. 1 <L. Venu [th, 6_5i_ 6 46 30-0 53_:7 36 -i 33 •• 2 9 _6_-9 42 -9 rjr Epheraerides at Transit [To/aap. iU. of the name of the next hour. 590. Accimitely.— Employ the decimals of the diff. for 10'" as whole seconds, taking care to divide the prop, part corresponding by 10, or by 100. Proceed as above directed in No. 589, (1 ) and (3); also take the seconds of the Greenwich Date as minutes, taking care to put the minutes of the prop, part into the place of the seconds. SUBOUniNATK COMrUTATIOXS. 213 tiTid the seconds into that of thirds : it is near enough to work to the fniction of 1"". I7'"38"'20": I Ex.2.Greon Date,Jaii.22.l, 1878, 4'' 3'" 45' '• find the Mnons (iirlin. .D. l3o"-28' Deol.4\ o''i5'27'7N.(/cc-. I IP": 170" 92 "3! a' " -9 2 7 I Rkd. Dkci,. 6 25 o N. Kx. I. Greon. Date, Aug. 16, 18 find the Muon's declin. ])..-l. I7^ 8° 10' 3"-2N.! 10'": i3o"-28::3Si'" +819-4 Uku. Dkci. 8~i8 22 6 N. The greatest change of docl. in 1 hour is 17' ; hence, to obtain tlie dcHjl. in the extreme case, true to I', the Greenwicyi Date must lio true to -f", or 1° of long. ; and to obtain it to 1", the Qreeuwich Data nuist be true to 4", in the extreme case. [8.] The Uoon't Right Ancennon. 691. Take the dlff. of R.A. for l\ To the const. 9-5229 add the prop. log. of the diff. for 1'', and the prop. log. of the minutes and Beeonds of the Greenwich Date: the sum is the prop. log. of the proportional part, always additive. When the sura exceeds 24\ reject 24''. E«. I. I 1' 17- IS': R.A. i\ It. A. the Moon's It. A. 17 560 11 ^1° 33-3 45 3 21 11 18 6 1878, E.. 2. G een . Date. Aiiril 28 h, 1878 ■ 6" 56- 45-. Hnd the Moon's H.A II. A. 16", 13" 58- 54-6 ■^229 '7'. 40 4 9 5229 9690 V.ir. i", • 45-8 20089 o'«5 Tim.., 56 45 ■5013 ■5104 U.A. 16", I 40 13 58 54-6 20331 RtD. R A. 34 -6 Tlie greatest change in P is 2™ 55', the smallest is 1"" 45' ; hence have the result true to 1', the Greenwich Date must be true to 20'. [9.] Right .592. M'ith the Green. Date and daily variation of R.A. deduce the prop, part by Table 21 ; this is to be mhlod to the R.A. at the preceding noon when incrennivg, and suhtrttcli'd when dwreasiafi. l-.x. I. Green. Unte, Sep I" 47» : find Venus' R.A. R.A. Sept. nth, Sept. I2lh, Didly Var. n'' 30-", var. 2'" O' 5-5 17'", 2 5-5 R.A. .Sept. nth. 1. nth. 1903, „h 37. iS-8 II 35 ■3 '3 2 57-5 I -4 II 37 18 -8 II 36 17 -5 Ex. a. Green. Oat-, May 5th, 1903, 22'^47'": find Venus' R..\. 5'' 17'" 49" '3 5 22 58 -6 5 9-3 R.A. May 5.h, May 6th, Daily Var. 22" 30"', var. 5'" May 5.1,. Hm,. It. a. 4 53-2 5 '7 49-3 5 22 42 -S The greatest daily change of K A. is G" i}14 NAUTICAL ABTRONOMT. [10.] DecHnation of Venus. i>93. Find the proportional part, and apply it to the <lecliii. at the preceding noon, as directed in No. 580. As the process, wliether Approximate or Accurate, is the same as that for the sun, no example is necessary. The greatest daily change of declination is 35', [11.] lUghi Ascetision and Declination of Mart. 594. Proceed as for Venus. The greatest daily change of R.A. ie 4"°; that of declination, 25', [12.] Right Ascension and Declination of Jupiter. 595. Proceed as for Venus. The greatest daily change of 11. A. is 1""; that of declination, 4'. [13.] Right Ascension and Declination of Saturn. 596. Proceed as for Venus. The greatest daily change of R.A. is 40' ; that of declination, 2'. 2. Reduction by Logarithms. 597. (1.) The proportional part may be found by the Propor- tional Logarithms, Table 74, thus: — For 24-hourly variations take the constant log. 91249; for 12-hourly variations take 8-8239; for 3-hourly variations, no constant; and for hourly variations, 95229. Then to the constant add the prop. log. of the Green. Date, (reading hours and min. as min. and sec. wlien the var. corresponds to more than 3''), and the prop. log. of the variation as given for 24'', 12'', S"", or P; the sum is the prop. log. of the proportional part required. Ex. 1. (Daily Variation.) Green. Tim 1 1** 30™, Daily Var. 14.' 42". const, log. 9' 1249 Gr.Time 11'' 30"' p. log. 1-1946 Var. 14' 42" P- log. 1-0880 Prop. Part 7' 2"'6 p. log. 1-4075 Ex. 2. (Twelve-hourly V.-.r.) Green Time 4'' 11™, Var. i6"-6. const, log. 88239 Gr. Time 4'' 1 1" p. log. 1-6337 Var. i6"-6 p. log. 2-8133 Prop. Part 5"-8 p. log. 3-2709 Ex.3. (Three-hourly Var.) Green. Time yh igni J2,', change in 3 hours I°3r4i": find the Prop. Part for 1'' iS™ 12'. Gr. Time i"" iS'" 12" p. log. 3621 Var. i°3i'4i" p. log. 1930 Prop. Part 0° 39' 49" p. log. 6551 Ex. 4. (Hourly A'^ar.) Green. Tini« 10'' 56'" 10", Hourly Var. 8' 47"-2. const, log. 9-5529 Gr. Time 56'" io» p. log. -5058 Var. 8'47"-2 p log. 1-3114 Prop. Part 8' i3"-5 p. log. 1-3401 (2.) The proportional part for 24'' is obtained conveniently fron> Table 21 A;* thus: — • In common practice at sea the prop, part may be taken out at sight from Table 21 : when extreme precision is required the logarithms to four places only are not .sufticient. For ex., at sea, for the Time 7'' lO'", and Daily Variation '22' 27"-5, we enter the table with 22' 30", and take out at once (No. 50) the quantity about i between 6' 33 '-? at 7'' 0"', and 7' V-g at 7'' 30'", that is, 6' 40", or 6'-7. Now this mental interpolation is performed in very considerably less time than it takes to write down the quantities, while the small inac- curacy to which it is liable, amounting here to 6' 42 "-4 — 6' 40 ', or 2"-4 only, would he wholly inapjireciable in practice at sea. The logarithms in Table 21 A give in this case the result true to 0"-l; but if the prop, pari were above 8' the logs, could no longer be Upended Ex-. 3. (The K<|UntioM Date, June 25th, 6i' of Time) Green Gr. Time 6" 1 1- Daily Var. 12- loR. i;890 log. 3010 Prop. Part 3' log. 8900 Kx. 4. (Right Ascension of Venuk.) Green. M.T. 19K i-i". Daily Var. 4"' 54*. Gr.Time 19' 'V" Daily Var. 4"' 54- Zl Prop. Part 3- 55* 7X6, 8UBORl)IN.\TF. C0.MPUT.\T10NS. 215 Tako out from tliis Table the lo>r. of the Greenwich Time, and lidd to it tlie log. of tiie Daily Variation ; tlie sum is the log-, of ll)oi prop, jiart required. Ex. 1. (The Sun's Declination.) Gr< Date, M.-iy 13th, 11" 30'". Gr.Time iiii 30" log. 3195 Daily Var. 14' 41" log. 2129 Prop. Part 7' i"-6 log. 5324 Ex. 2. (The Sun's Riglit Ascensii Grter- Date, June 6th, 9'' 19'". Gr. Time 9*" 19^" log. 4^<^9 Daily Var. 4" 7'-5 lug. 7648 Proi. Part 1'" 36* log. 1-1757 3. Correction for Second Differences. 598. The quantities in the Nautical Almanac do not in geneial change uniformly, that is, by equal portions in equal times, but the differences of any scries of quantities taken in order exhibit differ- ences among themselves, or second differences, as in the case of alts., }i. 200. Ht-nce the proportional part found by the preceding rules is not always the actual change in the interval, but may require a correction, which is called the equation of second differences. The greatest error which can arise in any case from neglecting this correction, that is, the greatest value of the equation itself, is J of the whole 2d diff. ; this takes place when the interval for which the proportional part is required is half the interval for which the quantities are set down in the table. For example, suppose ttie second diff. of the sun's decl. to be 26" in 24'' ; the greatest error of neglecting the equation will be i-8th of 26", and will take place when the Green. Date is I2i', or midnight. 599. To find the Equation of Second Differences. Take the two quantities in the table ne.xt on each side of the given one, and set them down in order. Add together the 1st and 4tli, and the 2d aiul 3d ; write against the sum of the 2d and 3d, whether it be the greater or the lesser of the two sums. Half the diff. of these two sums is the 2d diff. Under the Tabular Interval, and with the Green. Date as inter- mediate time, enter Table 25 and take out the multiplier, by which multiply the 2d diff. ; this is the Equation of 2d differences. If tlie 2d sum is marked the greater, add the equation to the prop, part deduced by one of the preceding rules; if the lesser, subtract the equation. opnn u iheaing the true tenth, not only because the last figure ceases to change by lat. 7'' .tH'«, but because the last figure of any logarithm is itself but an approximation. Altliough logarithms afford material bervice in multiplication or division of many figures, yet in short and easy reductions they are attended, as is well known to experienced arith- meticiani^, with considerable loss of time, and should accordingly be resorted to only when they nwequivocally effect a saving of time and labour. It ia also important to observe that the facility of mental interpolntion constancy impro»<6 bf czeiciie, and that the habit bharpens the ucrcej^liun of arithmetical proportiviu. 21G NAUTICAL ASTKONOMY. ]3y Logarithms. To the prop. log. of thn 2(1 difF. add the ar. co. log. of the multiplier ; the sum is the prop. log. of the equation required. Ex. Greenw'oh Date, June lytli, 1878, if ii"' M.T.: find the Sun's Declination. Hie two declinations preceding are those of the i6th and 17th ; the two following are tboseof the iSthand 19th. June .6th, 23° 22 3'- 5 In Table 25, Tabular Interval 24" and 17th, 23 23 57 "3 13" J I'" give -124. igth, 23 25 26 3 This multiplied by 2475 gives f'oj, 19th, 23 26 30 -6 the EauATioN of 2d Dikfs.; which being 46 48 added to prop, part as found by No. 580, 46 40 23 6(y ■ealer) gives Declin. required. By Logs. id Diff. ^4 75 24'-75 p. log. 2-6400 Log. 9-0925 ar. CO. 0-9075 P- log- 3-5+75 600. This correction is of the most importance when the quantity attains its mtixlmum, that is, arrives at its greatest amount between two times given in the Nautical Almanac. This circumstance is known thus : — When the sum of the vars. in 1 hour opposite the Green, day and the following one is equal to the diff. of the vars. in 1 horn' oppo- site the Greenwich day and the preceding one ; for ex. on Dec. 20th, 21st, and 22d, the vars. in 1 horn- of the sun are l"-70, 0"-52, and 0''-66 respectively, lience the declin. is maximum at some time lietween the noons of the 21st and 22d. III. Conversion of Times. 1 . Intervals. [1.] To convert an Interral of Mean Time into an Interval of Sidereal Time. 601. Appro.vimately. — Increase the Intei-val by l'" fur every 6 hours, or by 10' for each liour, or by P for every 6'". 602. Accurately. — Add to the Interval the Acceleration (Table 2.S), corresponding to the liours, minutes, and seconds. Ex. 2. (Accurately.) The same ex. 'ntebv. in Si3. r. .-97 6' -or \ Interv. in Sid. T. [2.] To convert an Interral of Siderer,. Time into an Intrrnal of Mean Time. 603. Approximately. — Diminish the Interval by 1"" for every 6 hours, or by 10' for each hour, or by 1' for every 6"*. CO-i. AcciLrately. — Subtract from the Interval the Retardation* (Table 24), corresponding to the hours, minutes, and seconds. * Or from corr. spouding tubles iu N.viit. Abniinac. SUBORDINATE COMPUTATIOXS. 217 (Approximnt ■ly.) Conv urt E (Accurate!; .? The sanie S. r. i„to .M. r. 7" 13- '7- 7" >^'S-8i] /'i3-'7" o- au.l 2- - I 12 13'" -I 10-99 s M.T. 7 12" s U 17* . IN M.T. 7^12 6 -01 The above precepts relate to Intervals of time; the following are employed in the conversion of absolute time of one kind into that of another. [1- 2. Absolute Times. mvert Apparent Time into Mean Time. 005. Reduce the Equation of Time, taken from page I. of the Nautical Almanac, or from Table 62 by No. .583, or 584, and apply it to the given App. Time as directed in the said p;ige I. or in Table 62. If the Eq. of T. when subtractive exceeds the A.T., add 24'' to the A.T. and date the time on the day before. F.x. 1. March 2H, 1902, at 11'' 56™ 43" M.. A.T., long. 148° W. : tind .M.T. The (^reen. Rite is 2' 9" 49'". E(). T. 2J, i2"27«-8 3 1, '2 1 5 -6 Daily \'ar. 12 2 g* 49'". var. i2"-2 -5 -i 2il, 12 27 -8 lie I. Fq. T. +12 22 7 App. T. 23 56 43_ iAlEAN Time, 2(1 09 57 Ex. 2. Xov. 10, 1902, o'' 13"' 40' r.»i., A.T., long. 36° E : re,iuire.i .M.T. Green. Date, <f 21*' 50'". Eq. T. 9th, -16"' 7" '6 loih, - 15 2 7 Daily Var. 4-9 21'' 50'", var. 4'-9 — 5 'O 9th, ^16 7 -6^ - 16 2^6 App. T. o 13 40 -o .Mean Timf, 9th 23 57 37 -4 [■2.] To convert Mean Time into Appiirent Time. 606. Find the Green. Date ; reduce to it the Eq. of T. from pnge 11. of the Nautical Almanac, or from Table 62, and apply it to tiie given .M.T. as directed in the said page II., or the contrary way to that directed in Table 62. If the Y.q of T. when subtractive exceeds the M.T., <idd 24'' to the :M T. and date the time on the day before. \v.. Ex. 2. Fell. 17th, igo2, long. I2o° E., o' 5'" iS- M.T.: Hnd A.T. Gn.ii. Date. :M.T., i6' i6'' 5 '• \;\. 1. Ausr. 31s', 1902 long. 18° 20'' 58'" sr yi:Y.: Kiul A.T. (iicen. Dale, ;M.T., 31'' 22'' IT". K..I. E.|. T. -o"II-I ."M.T, 3i^t, 20 58 51 -o .\n-. Ti.MK 31st 20 58 39 -9 [^] Toa.,n:rti That is. having given the Right Ascension of the Meridian, to find .Mean Time. 607. In W. long, add the Acceleration for the long, to the Sid.T. at mean noon ; in E. long, buhtract it. From the given Sid. Time (increased if necessary by 24 ) sub- }i(lercal Time into Mta 218 NAUTICAL ASTRONOMY. tract this rt-duced Sid. T. at tlie preceding noon ; the renin'.. ider is tlie approximate M. T. ; subtract from tliis time the Retardation corresponding (Table 24). El. 1. Jan. i.t. 1878, Sid. T. M. Noon, Aocel. 9" i"-i8«-7) 50- 8 -z 40- m) long. 9" so" 40' i8i'4+"'o-7 - I 37 -o E , at ..^9-^3'Sid.T.: Given Sid. T. Red. Sid. T. M. N Appiox M.T. „„._.. ,,.j| 59' oz) Mean Time, find M.T. Ill" 9ini5"o 18 41 23-7 1 26 59 -3 Sid. T. M. Noon, IS 4^ 23-7 —24-2 2 16 J51 Ex. 2. March I2d, 1878, long. 7'' ai">35' W., at Iji' s"'!?-! Sid. T. : find M.T. The Rkd Sid. T. is o'' o"'37"'9 ; whence the appiox. M.T. is 1 1''4'"49'-3, and the Ret. to '■9 sub. leaves Mean Tii ,h 3„, [4.] To convert Mean Time into Sidereal Time. That is, having given tlie Mean Time, to find the R.A. of the Meridian. 608. In W. long, add the Acceleration for the long, to the Sid. T. at the preceding mean noon; in E. long, subtract it. To this reduced Sid. T. at mean noon add the given M. T. and the Acceleration for the said M.T. ; the result (rejecting 24'' if it e.xceed 24'') is the Sid. T. required. Ex. 1. June i9th, iS-g, long, lo' jg" 6* I-., at -j" ■i7'" 46'-6 M.T. : find Sid. T. Sid. T. at M. Noon, igfh. 6'' 29'"44'-3 Accel, for long, lo'' 39"' 6" + i 4S "o Red. S.T. M. Noon, 6 31 29-3 M.T. 3 37 46 -6 Accel. 3'' 29'-6 ) 37"' 6-, +35-8 47' •■' Sid. Time, 10 9 51 7 Ex. 2. Nov. 26tli, 1878, long. 8'> 51" ic" E., at 141" 55'" 7-8 M.T. ; find S.T. Sid. T. M. Noon. 26th, 16" 21" 7'-6 Accel for 8'' 52'" 15' — i 27 -4 Red. S.T. at M. Noon, 16 19 40 z M.T. ,4 55 7-8 Accel. 14'' 2>" iS'-o \ 55"' 9'^ +2 27-0 7-8 -o) Sid Time, IV. Houk-Angles. 1. To find the Hour-angle, Mean Time being givm. [1.] Hour-angle of the Sun. 609. Find the Green. Date; Reduce to it the Eq. of T., and apply it to the M.T. as directed p;ige II. of the Nautical Almanac, or the contrary way to that directed in Table 62 ; the result is A-T. If A.T. is less than 12'', it is the Sun's Hour-angle, reckoning from the meridian westwards ; if A.T. exceed 12'', subtract it from 24" : the remainder is t!ie Hour-angle, reckoning from the meridian pR.st wards Sl'BORUlNATIC COMPUTATIONS. 2ir> B». 1. M«y 19*1'- 1878. long- 57° 4' '^^'•. tt 3" 7" 46* M.T.: find the Sun's Hour. The Green. Onte is 19" 6" 56" 2'. Eq. T. 19th, rage II. + 3'"45"3 loili, +3 4^5 I 8 6' 56", var. 1-8 -8 Rud. Eq. T. M.T HODR-AMOLK, 3 4S-3 + 3 44 5 3 7 46 o 3 II 30-5 Ki. a. July 2d, 1878, Icng. 6 at 20" 26" 53* M.T.: find the Sun ant'le. Tlie Green. Date is i' iS'' i8" Eq. T. 2ii, Page II. 3" 3-', _3_ Daily Var. 16'' 19"", var. ii«-2 Suh. fioin M.T. M.T. 43 9 5. -6 53 o 20 23 3 30 610. \V hen tlie Sun'.s Hour-aii_>fIe is required from iiiidiiiglit, if A.T. is less than 12'', subtract it from 12'*; the remainder is tlie Hour-angle, reckoned westwards. If A.T. exceed 12'', subtract 12' from it; tiie remainder is the Hour-angle, reckoned eastwards. [2.] Hour-angle of a Star. 611. (1.) Find the Green. Date, to which reduce the Sid. T. at mean noon. (2.) To the M.T. add this reduced Sid. T., and from the sum (increased if necessary by 24'') subtract the star's R.A. ; the result is tlie Hour-angle \V. If the Hour-angle exceed 12*", subtract it from 24''; the remainder is the Hour-angle K Ej. I. July 2ist. 1878, long 32° 10' W., Ex. 2. Sept. 1st, 1878, long. 169^ 57' E. »t 9' 45" 2i» Rl.r.: required tlie Hour- at 8" 57" 39' M.T. : find the Hour anfle »ngle of Arcturus. ofAltair. Green. Date, ii'' 11'' 54" iV Green. Date, Aug. 31'' 11'' 37" 51". Sid. T. Mean Noon, ziNt, 7" 56'" 28-5 Sid. T. at M. Noon, 31st, 10" 38" 7-3 Aceel. II", 54-. lied. Sid. T. 7 58 25 -8 9 45 ?> 17 43 46 8 Accel. 21', 3 27 37-, 6-> 51', I M.f. Red. Sid. T. 10 41 40 5 M.r. 8 57 390 * R.A. HoOX-iHOLI, 3 33 18-4 19 39 19 5 * R.A. -19 44 53 s 23 54 ^6oW. HOUK-..C..P, 5 34 oE. El. 3. Oct. iBt, 1 nf Markxh. Ei. 4. Dec. 25th 878,1 187S .ng. 92" 48' K.. a , l,M,g. 86- 4S- M t 5'' 58" 19' M.T. : required tlie Hour-angl* Hora-ANGLE, 4" 20" 7-8 E. ■., a, 5^ r 35- MT.: find Higels Hour- .ngle. F.x. 5. Alarcli %n of Antares. , i«78, long. 110° 39' V lIoUK-A.NcLi.:, 5" 43'" 55* E. v., ai 11" 3'" 37- M.T.: find the H..nr.a„gl« H0UH-*II0I.E, s<- 15- 53 9- 1.. [3.] Honr-anrjle of a Planet or the Moon. 612. (1.) Find the Green. Date, and reduce thento the Sid. T. at moan noon, and the Il.A. of the body. (2.) Add this reduced SiA T. to the M.T., and proceefj ss for a star. 220 NAUTICAL ASTKOXOMT. lOx. I. Oct. I5tli, igygjong. 41° 44' W. It 6" $6" 54« r..M. BI.T. : find the Sloon' Hour-angle. Green. Date, Oct, 15" 9*" 43" 50', Sid. T. Mean Nuon, 15th, 13'' 35" jz'z Accel, g'' 43° so- li ert Sid. T. H.A. q" t s R.A. 9" lied. R.A. lied. Sid. T. w. r. f. li.A. Iio0K-* 287 13 37 81 4 40" 20-5 4 4^ 37-5 952^9 2 17 1-8967 43 50 0-6.35 I 39 9 20331 4 4° 2° '5 4 42 0-4 ,31, 37„, g.., 6 56 54 20 34 2-1 -4 42 0-4 15 52 i-7\V. Ex. 2. Fell, nth, 1S78, loi.s 87° 6' W., at 4'' 46" 48' A.M. iM.r. : tiiid the Hi.ur- angle of Mars. Green. Date, Feh. 10'' 22'' 35" i2». Sid. T. Mean Noon, 101 h 21" 21" 43-0 Aciel. 22" 3 36 -8 35° Si Hed. Sid. T. 21 15 is -6 Mars' R..\. 10th i"" i5'"57'-o Daily Var. 35" R.A. loth Red. R.A Red. Sid. T ;m.t. Mars' R.A. » 25-2 ^5• gives 21" 25" 25-5 4 46 48 26 1213 6 - 2 18 14-1 13__53__59 5 "'• o 6 o -5 E. 8 7 58'3E. 2. To fnd the Honr-avgle, the AUitvde being given. 613. By Inspection. See Explan. of Table 5. 614. By Computation. Atld togetlier the alt., lat., ami pol. dist., take half the sum, and from it subtract the alt. Add together the log. sec. of the lat., the log. cosec. of the ]iol. dist., the log. cos. of the half sum, and the log. sine of the remainder; the sum (rejecting tens) is the log. sine square of the Hour-angle.* of the logarithms give it to the Ex. 1. Alt. 37° 51', lat. ;i° 10' N., pel. dist. 70° 33', or decl. ig° 27' N. : find the Hour-angle. See Ex. 1, of No. 615. Alt. 37° 5"' Lat. 51 10 . . sec. 0-20269 Pol. dist. 70 33 ... cose( Sum 159 34 Half 79 47 ... COS. Rem. 41 56 ... sin. Hora- ANiiLE 3'' 31"' 47' sin. sq. =2552 9-24888 9"8249S 9"30204 E.\-. 2. Alt. 21° 19' 5", lat. 51° 9' 26" N. docl. 11° 14' 44" S. : find the Hour-angle. Alt. 21° 19' 5" Pts. for' Lat. 51 9 26 ...sec. 0-202536, -(■ 78 P.dist. loi 14 44 .. cosec. 0008414, -H 6 «77 173 43 >5 S6 5. 37.. 65 32 32 •■ cos. 8-738810, sin. 9-9 59 '67, 8-908937 — 201 ,. in. sq. 8-908736 2'" 19' 707 -3 29 i HOUR-AN-GI-E 2 12 I9-3 Ex. 3. Lat. 30^ ir24"N. Dcd. 14° 2' 46" N. Alt. 61=9' 17". Hour-angle il- 43™ 52'. When both the lat. and decl. are 0, the zenith distance in time is the measure of the Hour-angle. At sea it is near enough to take the alt., lat., and pol. dist., to the nearest minute; but if the sum is odd and greater that 170°, take the cos. and sin. to 30", because the neglect of this may make a sensible error in the Hour-angle. Log. I square. Table 69, is the same as the log. havrrs; 8UBORDINATK COMPrTATlONS, 221 [1.] Errors of I Zip Hour- Angle. 615. The following- rules give, very nearly, the effect of 1' error ill tie alt., hit., and pol. dist., and therefore for any small number of niin. or sec. in the like proportion : — (I.) Error of hour-anirle, or time, due to 1' error of alt.* Add together the parts for :iO" of the cos. and sine : the sum, divided by the parts for 1» (Tab. 69), gives the error required. When the alt. is too small, the hour-angle is too great; when the alt. is too great, the hour-angle is too small. (2.) Error of hour-angle, or lime, due to 1' error of lat.f Miil- tiplv the parts for 30" of the sec. by 2, and add the parts for the sine'; under the sum ]3ut the parts for 30" of the cos., and take the dirt".; divide this ditf. by the parts for 1'. When the lat. and true bearing are of the same names, the errors of the hour-angle and lat. are of the AY(/He kind ; when of contrary ;ianies, oi' contrary kinds. Ex In N. Lat., if tlie sun is to the N. of E. or W., and the Lat. employed is too great, the computed hour-angle wiU be too great; if the sun is to the S., in the same case too gmalt. (3.) Error of time, or hour-angle, due to 1' error of pol. dist. Multiply the parts for 30" of the cosec. by 2, and add the i)arts for 30" of the cos. ; under the sum put the parts for 30" of the sine; take the ditf., and divide it by the parts for 1'. \M)en the parts for 30" of the sine are less than the sum over them, the error of the hour-angle is of the contrary kind to that of the pol. dist. ; when greater, of the same kind. Ex. See Ex. 1, of No. 614. 51- 10 sec. 70 33 cose( 79 47 cos. 41 ;6 sin. • UM. 69 1 6 p. 83U J ^ Error 1 of AIL Cos. 354 Sin. 7 1 (Sum) 4"^ Error of Time 64 ■■T Sin. (Sum) .56 227 Cos. 3^ (i)itr.) .27 Error of Time of Pol. Dist. Cos. 354 (Sum) 398 Sin. 71 (Diff.) JI^ Error op Timj 3^7 The error of the hour-angle may, possibly, be made up of the sum of these three errors, but in most cases they will partially compensate. * To find, appro.Timately, the small interval of time corresponding to a small change of ■It. by means of the Azimuth: — .\dd together the log. sine of the change of alt., tlie log. cosec. of the azim., and the log. sec. of the lat. : the sum (rejecting tens) is the log. sine of the interval re€|uired. To find the same, by means of the Hour-angle : — Add together the log. sine of the change of alt., the log. sec. of the lat. and declin., the log. cos. of the alt., and tlie log. cosec. of tlie hour-angle : the sum is the log. sine, as above. One of these processes may, on some occasions, be convenient. + To find this error by means of the Azimuth : — Add together the log. cot. of the azim.. tlie log. sec. of the alt., and the log. sine of the error of lat. : the sum is the log. sine ot tlie 222 NAUTICAL ASTRONOMY. 3. To find the Hour-angle, the Azimuth hciny given. fil6. Add together the log. sine of the aziiniith, the log. cos. of the lat., and the log. sec. of the decl. ; the sum (rejecting teasy is the log. sine of the angle A.* Under A put the azimuth, reckoned from the elevated pole, and take half the sum. Take half the sum of the pol. dist. and colat., and half the ditf. Add together the log. tan. of the half sum of A and the azim., the log. cos. of the half sum of the p. dist. and colat., arid the log. sec. of the halfdiff. ; the sum (rejecting tens) is the log. cot. of an arc. When each half sum is less, or greater, than 90*', twice this arc ifi the Hour-angle required; but if one only of the half sums exceed 90°, twice liie suppl. of the are is the Hour-angle. Ex. Lat. 51° 30' N., decl. 20° i' N., azira. N. 110° ii' W. • find the Hour-angle. Az. IIO°2.' sin. 9-97201 P. Dist. 69° 58' Lat. 51 30 COS. 9-794'5 Colat. 38 30 74° 23' tan. ,i359 t)ecl. 20 2 A 38 25 Az. no zi Sum 148 46. sin. 9-793^7 half 74° 23' •Sum Diff. '3. 28 half 54 ,4 do. 15 44 COS. 9-7'i677 cot. 03 3694 4. To find the Hour-angle, the Altitude and Azimuth being given. 617. Add together the log. sine of the azim., the log. cos. of the alt., and the log. sec. of the decl. ; the sum (rejecting tens) is the log. sine of the Hour-angle. Ex. Alt. 40""' 25', azim. 69° 39', decl. 20^ 2': required tlie Hour-aiigU« Az. 69° 39' sin. 9'9720i Alt. 40 25 COS. 9-S«.5g Decl. 20 2 sec. 0-02711 GLE, 3" 17"' 48' sin. 9 88070 5. To find the Hour-angle on the Prime Vertical. 618. Bg Inspection. See Table 29. 619. Sg Computation. Add together the log. cot. of the )at. and the log. tan. of the decl. ; the sum (rejecting tens) is the log. COS. of the Hour-angle. find the Hour-angle of a cel««tiAl Ex. Lat. 31° 28', Decl. 14° 11' of the same name: find yo the prime vertical. Lat. 31° 28' cot. 0-21325 Decl. 14 II tan. 9-4026? HODR-.^NGLE, 41' 22'" 26' COS. '^I'^l 6. To find the Hour-angle at Rising or Setting. 620. Bg Inspection. When the decl. is less than 24°, take out ol • This inRlc A is the angle at the body contained between its pol. dist. and len. diit.. cf ujle PAZ, (ig. p 162. SCBORDIXATK COMl'UTATIONS. 22;? Tiifcle 26 the time of settin// ; this is the Hour-angle required. If is called also the Semidiurnal arc. When the dec!, exceeds 24", see No. 621, or Explan. of Table T) 621. Bi/ Computation. Add together the log. tangents of the lat. and dccl. ; the sum (rejecthig tens) is the log. cos. of the Hour- angle at rising or setting, or its supplement. \\'hen the lat. and deelin. are of the same name, take the supple, nicitl ; when oi contrary names the Hour-angle is that taken out. Ex.1. Lat. 48° 42' R deol. 20° 1 1' N. find tLe Hour-angle at rising or setting. Lat. 48° 4J' tan. 0-0562 Decl. 20 11 tan. 9' 56 54 1-6216 MuCR- 58 56 E.X. 2. find the H Lat. 3i°io'N. d ur-angle at rising eel. ..°I4'S ir setting. Lat. Decl. Hour-angle, 31° 10' II 14 5" 32". 24- tan tan 9-29«o 9-0796 7. To find the Hour-angle near the Meridian, hy the observed Change of Altitude. 622. The alts, must be on the same side of the meridian. Correct the diff. of alts, and the interval by adding the correction the following table : — Ti «E. i c 43"' '5' V 0' 0' 0" 6°15' o' 44" 10' 45' 3' 5'" 13 44 .6 30 , 30 50 11 4 7 ■20 45 2 2 45 056 4 ^5 23 46 30 3 7 I 3 4 43 25 47 45 4 15 26 48 3 5 30 I 17 12 5 ^' 28 49 15 6 45 I 25 5 4^ 30 50 30 8 8 ' 3+ 32 51 45 10 15 I 44 6 27 33 52 4 12 30 > 53 13 6 51 34 53 15 '4 45 2 3 35 54 30 >7 9 2 14 7 4> 36 55 45 15 2 26 37 58 5 ^3 30 2 39 14 8 36 38 .■)7 15 ^7 45 2 52 40 58 30 Q 10 3 >6 9 34 41 59 45 55 15 3 20 42 H 60 C ° 39 30 3 34 15 10 37 Add together the log. sin. of the diff. alts, (thus corrected), the log. cosec. of the interval (corrected), the log. sec. of the deelin., the log. cos. of the mean of the two alts., and the log. sec. of the lat.: the sum (rejecting tens) is the log. sine of the hour-angle at the middle of the interval, nearly. To find the hour-angle for the alt. nearest the meridian, subtract half the interval from this hour-angle. To find the hour-angle for (lie nh. furthest from the meridian, ofW half the interval to the hour- Hn<jle found. Aote.—l( the alts, are not ine,i.sn 8gure.s No. 4.52, may he en\plnyed, recii 3ire|il when helow the [pole, when it is I rid. alt., de<luced from the lat. hy aec this alt. is always Bomewhat too great. 224 rrATTTICAL ASTnOX().\rY. -at. 51° 30' N., (Ircl. 22° 20' N., alts. 60° 27'52" anil 60° 34' 35", 6' 43" at an interval iif 4'" : find inprlu at ilic time of the alt. Alt. 60 31 51 30 Int. o'' 21'" 58" -2 NGLE 19 58 no err.) sin. 7-2909 (do ) cosec. 175S1 c. 0-0339 9-6921 02058 Ex. 2. Lat 40° N. dccl 20° N., ob. t„i„ed tr. alts. 69° 5S' and 67° 0', or diC alt. 2" 58', with inlcrv. ..t47"'39'- findtlie H..ur-aiigle at llie time of the alt. furtliest from the meridian. D. Alt. 2° 58' 0" Int. 47- 39- Corr. +5 Corr. +21 2 58 5 48 0' D. All. 2° 58' 5" sin. S-7142 Int. 48- o- cotec. 0-6821 IXcl. 20° 0' sec. 0-0270 !\lean Alt. 68 29 COS. 9-5644 Lat. 40 sec. J)-II57 Mid. T. 29-" I o- sin. 9-1034 H.R-ANGLE 52 59 (only 2' too small.) The degree of depeadence is chiefly to be estimated from the effect produced by a small change in the difif. alts. For finding by an easy operation the ap])arent local time from an observed altitude, Davis's "Chronometer' Tables (J. D. Potter, London, 10.<t. ^d ) will be found of service ; they also make clear the effect and direction of any small error in the observer's latitude. V. Times of certain Phenomena. 1. Time, of Passitig the Meridian. [1.] Mtrldian Passage of tkt Sun. G23. The Apparent Time of the sun's meridian passage is 0'' 0™ 0* except below the pole, when it is 12'' 0" 0^. C24. To find the Mean Time of the meridian passage : — Take the Eq. of T. from page I. of the Nautical Almanac, or from Table 62 ; reduce it for the long, as the Green. Date. Then, if the reduced Eq. of T. is additive to A.T., it is the time P.M. of the sun's meridian passage. If the Eq. of T. be subtractive from A.T., subtract it from 12'' : the remainder is the M.T. of passage. Ex. I. Marcl) 31st, 1902, long. 140° W.: find Mean Time of Sun's meridian passage. Eq. T. 31<t, +4'- 28" -3 Daily Long. 9' lS-2 Ex. 2. Dec. 1st, 1902, long. 93° E. ; find Wean Time of Sun's meridian passage. Green. Date, Nov, 30'' 17'' 4S'". 7 -o 4_28 -3 Eii.T. „.Wto A.T. 4 21 -3 ' M. I'ass. 12'' 4'" 21' 3. [•_'.] Meridi,,,, Pnssaije of a Star. 025. To find the Apparent Time of a star's meridian passiige ; At Sea. — See Table 27, and Explanation. Eq. T. 30th, 31^'. Daily Var. 17'' 48'", var. 21- S -I I'" 29-1 -II 7-3 21 -8 -16 -4 lU-d. Eq. T. M. T. OK l^^ss. n '4S II 29 -I -li 12 7 47'-3" SUHnUDIN'ATE COMPUTATIONS. 225 Or, from the R.A. of the star (adding 24'' if necessary) subtract the R.A. of the sun at noon, Nautical Almanac, page I., or deduced from Sidereal Time in Table Gl (see Note, page 211); the remainder is the A.T. required. El. I. Oct. 17, 902: find A.T. of the Mer Pas.s. of Sirius. By Table 2 By Sun's R.A. Oct. 1st, IS- 14" R.A. Sirius 6" 41" For 17 (lays 59 Oct. 17th. O's R.A. 13 :6 Mer. Pass. 17 15 P.M. Mer. Pass. 17 1? Or iSth, 5 13 A.M. Or iSth, 5 13 Urs. M.^j., above and liplow llie Pole Ans. I" 19"- A.M.; I'' 17" p.' Ex. 2. Find the A.T. of the Jler. Pass, of Feb. nth, igo2. Ex. 3. Find A.T. of Mer. Pass, of Capella on July 20lh, 1902 Ans. 9" U" 9" 9-, 626. To find the Mean Time of a star's meridian passage: — Accurately. — From the R.A. of the star (increased, if necessary, by 24'') subtract the Sid. T. at mean noon on the day : the remainder is the approx. M.T. of transit. Subtract from this the Retardation, Table 24. In W. Long. siiUnict from this result the Acceleration for the Long. In E. l^ong. luld the Acceleration. The result is the M.T. of meridian passage. Ex. I. Jan. 1st, 1902, Ion;;. 1° 25' W. : find M.T. of Mer. Pass, of Aldebaran. R..\. Aldebaran 4'' 30"" 17' Sid. T. Mean Noon _40_48jS '9 49 29 "3 Ex. 2. May 22d, 1902, long. 131° It' E. : find JI.T. of Mer. Pass, of Spiea. 13I1 20" 4' -6 3 56 42 -6 9 23 22 o Ret. 9'' 23"< 22« —I 32 -6 9 21 49-4 Long. 8'' 44" 44' +1 26-2 M.T. OK Pass. '9~^3~ '5 6 Ret. 9" i'"28'7) 49™ S -o [ - I 36 -8 29- -I ) 1° 25' \V., or 5" 40" - -9 ALT. Mf.b. Pass. 9 47 51 -6 Ex. 3. Aug. 8tb, 1902, long. 90° 15' E. : find JLT. of Mer Pass, of Altair. .■\ns. 10'' 41'" 3'-4. Ex 4. Feb. 1st, 1902, long. 172° 34' W. : find .^LT. of Mer. Pass, of Regiilus. Ans. 13" 16'" 5'n. [3.] MeiUUun Panta^ of ihe Moon. 627. This is required only approximately. In W. Jyong. take from the Naut. Almanac the diff. between the Mer. Pass, of the propo.sed day and tlie next (given in mean time to ()'" )). In E. I>ong. take the diff'. between that for the proposed day and the day before. The diff. is the daily variation. Take from Table 28 the correction corresponding to the daily variation and longitude. In \V. Long, add this corr. to the time of 226 NAUTICAI, ASTRONOMY. mer. pass, on tlio given day; in E. Long, subtract it; the result is tlie time required. When one mer. pass, has 23^, and the next O*', 24" must be added to the hitter in finding the Daily Variation. F.I. I. Fin,! Mur. Pass 1878, lone. 46" W. Mer. Pass. i6tl., I7tli, of €. Jan. I .0" 9-1 n II 6 I 2 5 + 7-6 10 9 I 10 16 7 7 P.M. 6tli, E«. 2. July 24.11., 18; Hnd !lie Wlt. Pa«. of tilt Mer. Pass. Daily Var. 130° E. var. 47"-7 MEa. Pass. July 24tli, at 6"45» 8, long. 130° L Aloon. 23-19' 1-8 22 18 14 i Daily Var. 47 -7 46<'W. var. 61-5 Mkr. Pass, Jan. ibtli.at lo" 16- -16 -8 23 19 I -8 23 18 45 -o 628. As the lunar day, or the interval between the moon's mer. pass, and her return to the same meridian again, exceeds 24 hours or a mean solar day, an entire day passes at certaia intervals without a lunar transit. For ex. : — The moon passes the meridian on the 3d, at 23'' 50"", or 10™ before the noon concluding the 3d. The lunar day being, at least, 40"' longer than the mean solar day, the moon will not have reached the merid. by about 30'" at next lioon, or that concluding the 4th ; she accordingly passes the mend, about 0'' 30"' on the 5th, having skipped the 4th altogether There may thus be no mer. pass, on the day proposed.* Ex. 1. March 3rd, 187?, Snd the Moon's Mer. Pass. Mer. Pass. 2 D.iily V.ir. ,ong. 21° W. var. 39"-4 ng. 31° W. i3"44"-i ° »3 -5 39 4 Mer. Pa. March 3rJ at ' 46»- 1 13 46 Ex. 2. October 26th, 1878, long. 38" E.: find the Moon's Mer. Pass. Mer. Pass. 26'' o"" 7"7 25 * * 24 23 10 -2 Daily Var. 57 5 Long. 38° E., var. ST"5 5 7 »6 g 7 7 IMer. Pass. 26 o 2-0 October 26th, at o"" a," p.m. In W. Long., when the sum of the corr. and mer. j^ass. exceeds 24'', subtract 24'', and reckon tlie time on tlie next day. In E. long., when the corr. exceeds the time of mer. jjass., add 24'' to the latter, and reckon the time on the day before. Ex. 2. Suppose Ex. 2 above, the long to he 90° E. Long. 90° E., var. sT°-S - ' 3'°7 i6' o" 7 7 X. 1. Supp 3se Ex. 1 above, the 1 . .70° w. g. 170° w. va r. 39-4 +.8-0 2" 23 44 -I Mer. Pa «. 302 -I March yd. a O* 2"-I P.M. Mer. Pa: October 26tb, i''54"'/ 13 S4 * This occurs abont the time of conjunction with the sun, and the day skipped is marked <5 in the Nautical Almanac. In like manner a day is skipped at the loiver transit (under the. pole) at opposition. STTBORPINATE COMPUTATIONS. 227 [4.] Meridian Postage of a Planet. (J29. The nioridiiin passages of the planets, like those of the moon, are given in the Nautical Almanac to 0'"-l of mean time. A planet, of which the R.A. increases faster than that of the sun, ski] s a day at conjunction, as observed in No. 628 of the moon. On the other hand, when the R.A. diminishes, or the motion of the jilanet among the stars is reversed, two transits occur within the limits of the mean solar day. As the greatest daily variation of meridian passage of Venn* amounts to 6"" only, the mer. passages of the planets may be taken at once from the Nautical Almanac for all practical pui'posea. 2. Time of Passage of the Pnme VerticaL [1.] Of the Sun. 630. Approximatehj. Find the Ilonr-angle by Table 29 : this is the App. 'i'inie, approximately, of the afternoon passage; the supple- ment to 12'' is the Aj)prox. Appar. Time of the forenoon passage. Prin Ex. 1. Jan. 20th, 1S78, lat. 39° S. : tho times of the Sun's Paasage of the Ex. 2 find the ; Vertical. Lat. 55° (led. 23° 27' N., or 23I ', Hour, angle 4'' 52'", which is p.m. traii-sit : tti4 other passage is at ^*' 8" A.ii. Jan. 20th, Sun's Decl.20° 5' S., Table 29, lat. 39° and docl. 20°, give Hour-angle 4" 1 3"'. The AT. of the W. transit is 4" 1 3°' P.M., that of the E. is 12'' —4'' 13'", or V"" 47"' A-M- 631. Accurately. Having found the Approx. App. Time as abovo (No. 630), apply to it the long, in time; this gives the Green. Dato in Apj). Time. To this reduce the sun's declination, and compute the hour-angle by No. 619. Eic. I. Aug. »9th, 1878. required tlic App. Time of Pas'^agc P.M. at Tunby, in lat 51° 40 20" N., long. 4° 41' W. Part, for Ut. 5..r<lecl.9r-l TablL-iOcivcs / 5i''4o' 20-cnt. 9-89SOIO - 86 5'' -io" 9 14 22 tan. 9-21 lOiS +295 4^4,' W. + '9 91 0902 S Rrecn. Pat.. 29th, 5 49 + 209 Decl. 29ih, 9" '9 33"-8 N. Cos. 9-109237 30th. ^ 58 6 •. x\. Pa«. p. Vektical, s' 30" 17* 5 27 49 ■7 5 12 9 •9 33 ■X 0485 6640 9 14 11 8 N. \t. 2. May 13th, 1878, tind Ibe TiuM of Passage a.'h. at Soulli .Sliiflds, lal. 55° o' 50" N., long. 1° 25' W. Gtcen. Date, Mav 12'' iq''o'* Red. Dcclin. i8''22'i6'N. Arp. Time Pass. 6'' 53°' 45' a.m. [».] Qr« Star. 632. Find the A.T. of m.Tidian p.is.s.age. When the time of the ta»t transit is required, siilnract the Hour-alible (Tabic 2iiJ from 228 NAUTICAL ASTRONOMY. tliis A.T. (increased if necessary by 24''); for the time oi' went, ti'ansit. add the Hour-anele. Ex. 1 . Find the Times of Eastern and Western Transits of Prime Vertical of Aldsbsran «t So. SbieUls, on Jan. ist, 1878. App. Time Mer. Pass. Tab. 27 9" 4'" 9*4'" Decl. le'^lat. 55° - 5 14 J 14 ApP. TiMEOF. E.TaANSiT, 4 27 P.M. 14 f5 W. Transit of 2d, z 55 a.m. Ex. 2. July llth, 187S, lat. 51° 30' N. : find Times of E. and W. Transits of Prim« Vertical of <t Lyrse. Ans. App. T. of Pass. E. 7'' So" P M.; W. 2'' 30'" am. Ex. 3. Dec. 4tu, 1878, lat. 40° 10' S : find Times of E. and W. Transits of Prime Vertical of Aatares. Ans. App. T. of Pass. E S'i^a.m.; W. 3I' 17" p.m. Et. 4. Aug. 17th, 1878 lat. 56= 3' N. : find Time of E. Transit of Prime Vertical o( Allair. A7ta. App. T. OF Pass. E. 4'' 22"' p.m. [3.] Of the Moon. aXi. Appro.rhnatehj. Proceed as for a star, using M.T. for A.T., becaii.so tiie time of her mer. pass, is given in M.T. 634. More Accurateli/. Find the approximate time as for a star ; find the Green. Date, and reduce to it tiie decUnation. Find tiie Hour-angle by No. 619. This Hour-angle, witii the correct time of mer. passage, gives the time more nearly. Correct the declination and re]ieat the computation. For extreme precision, a correction would be required for the oblateness of the earth. [4.] Of a Planet. 63.5. Find the M.T. of the Meridian Passage of the planet, in the Nautical Ahnanac, and apply the Hour-angle as directed for a star; the result is in M.T. n. igtli, 1878. lat. 54° 33'S. : | Ex. 2. Aug. 9th, 187S, lat. 49° ?6' S.: . -' "^ '- ' '■ " • -' And the Time of E. Transit of Prime Ver- tical of Jupiter. M.T. Mer. Pass. 9th) j^,, ^^„, page 2i;4 N.A. ( Lat. 50° ,s., Decl 21''.'-'. —4 10 M.T. ut Pass. ITT: r.M of W. Transit of Prime V I uf Venus. M.T. Mer. Pass, 19th | page 2."4 N.A. ( Lat 54°S.,Decl. C S. M.T. OF Pass. 3 Tiincs of liisimj and Seltinq. Tiicse are required ap[)roximately only. [1.] Of the Hun. 636. See Table 26, and E.xplanatiun. [2.] Of a Star, the Moon, or a Planet. 637. Find tiie A.T. (or M.T., according as required) of the meridian passage, No. 625, &c. Find the Hour-angle at rising or setting. No. 620. To find the time o{ rising, subtract this Hour-angle from the time of mer. jiassage (increased if necessary by 24''); to find the Time of seltiny, add them together, rejecting 21'' if the sum exceed 24''. SLTBORDINATK fOMPHTATIONS. 2-'a Ex. Jnn. ist, 1S7S, i.T. Mcr. Pass., Table 1 J5"N., Oecl. 16° N. A.T. or Rising 50° N. : find A.T. of i iiii; anil 9.-tting of AMeba urn A.T. Me r. Pass. 9' 7 4>' 37 AT. OF Skttino • 7 i» Or at 5" I8'"A.M onzd. fi^8. To find tlie cliaiifre in the time of ajiparent rising or setting due to flio iiorizonfal refraction and tlie lieiglit of the spectator, No. 44(5 (I) and (-2). Jii/ Comjnitation. Add topjether the log. .secants of the latitude and d(X'liiiatioii, tlie log. cosec. of tlie lioiir-aiigle at rising or setting, Biid the log. sine of 34' + (le])r. for the height of the eye, Table 8 ; the sum is the log. sine of tlie portion of time required, nearly. Ex. 1. Find the difference of times ( Sunset to an eye at the level of the sea, an on the summit of the Peak of Teneriffe, o May 4th. Hour-angle at setting (No. 621), 61" 35"' 52 La*:. i8° i6' sec. o'055i Decl. 16 10 sec. 00175 Il.-Ang. 6'' 36"' cosec. 0-005+ 34'+ 1 17=2° 31' sin. 8-6.)26 Time reu. 12"' 3" sin. 87206 Ex. 2. Lat. 28° i6'N., dedin. 16° 10' N. : re(|uired the difference in I he times ol Sunset to the eye at the level of the sea, and elevated 16 feet above it. 1 1 our-angle at level of the sea, 6'' 35" 52'. Lat. sec. 0-0551 Decl. sec. 0-0175 Hour-angle cosec. 00054 34 -h 4', sin. 8-0435 Time reu. 3"" 2' sin. J-I2I5 This process is very nearly correct in low latitudes, but in high latitudes, where the body, instead of ra])idiy jiassing the horizon, partly skims along it, the result, when the dip is large, is too fimali. Thus, for the above depression, 117', in hit. .50° (and declination above), tlie time comes out 17'" 2;^', it should be 17" 38'; and in lat. 6()0 the re.sult, 24."- 17% should be 25™ 4'. 639. More accurately, find the Hour-angle of the given celestial body when below the horizon 34' 4- de])ression due to the observei's lieigiit, by No. 642; this is effected by using 34'-fdepr., instead ol 18°. The Uiff. between this Hour-angle and that found by No. 621 is the portion of time required.* 640. Since the moon's parallax exceeds the refraction, Nos. 433 and 436, she always ap])ears below her true place, and therefore ri.«es later, and sets earlier, than a more distant body of the same declination. Accordingly, in the preceding rule we must use, in- stead of 34', the diff. between the hor. j)ar. and 34', and the differ- ence instead of the sum of the latter and the depression. If the depression is the greater, the rising is accelerated, otherwise re- tarded. For the hor. ])ar. 61', these effects neutralise each oilier at the height of 6.50 feet; for .53', at 320 feet; that is, to the eye ,)laced at these heiglits the moon in these cases rises and sets nearly at her true time. strictness, however III oiriutiiran, iiuwrvci, suiiic Liirrci:iniii ibcn (he bod} is «een at a cunaiderablc drpressi (nubti active) is doe to the refmctioa it«I/ 230 NAUTICAL ASTRONOMY. 4. Tiniea of the Beginning and End of Twilight 641. By Inspection. See Explanation of Table 5. 642. By Computation. Add together 18°, tlie lat., and the pol. ^dist., take half the sum, and from it subtract 18°, or the upi)er term. Add together the log. sec. of the lat, the log. cosec. of the poL dist., the log. sine of the half sum, and the log. cos. of the re- mainder; the sum (rejecting tens) is the log. sine square of the nun's hour-angle when 18° below tlie horizon. This Hour-angle is the App. time of the end of twilight, p.m. ; and the supplement to 12'' is the App. time of the beginning, a.m. Note. — ^Tlie declination at noon, and 4, or even 3, places in the logs, are enough for this purpose. Ex.2. Dec. 2ist, 187S, lat. 55° i' N. : find the Beginning and End of Twilight. Const. 18° o' Ej. 1. April 22d, 1878, lat. 5i°46'N the Beginning and End of Twiliglit. Const I go Lat. 51 46 sec. 0-2084 F.D. 77 »47 15 31 cosec. 001 00 73 45 sine 9-9823 55 45 cosine 9-7504 End gi" 28°' sine sq. 9-9511 Lat. 55 I P.D. 113 27 1S6 28 see. 0-2416 cosec. o-o;74 93 '+ 75 >4 sine 9-9993 cosine 9-4063 sines<r.9 6S46 begins 2" 8"' a.m , ends 9H 5,n. P.M. begins, 6" 42". A. M., ends 5" 18'" p.M, Ex. 3. March 3d, 1878, lat. 60° 47' S. Twilight begins Ex.4. Jan. 2d, 1878, lat. 70° I'N., T the sun not appearing above tlie horizon. 64,3. The duration of twilight, or the interval between the begin- ning of twilight .and the sun's rising, or between sunset and darkness, is found by taking the diticreiices of these times. Tims, in Ex. 1, it is 9" 28'" - T*- 3'" (setting. Table 26), or 4" 57'" (rising) -2" 32'", which is 2" 2-5". In Ex. 2,''it is 5" 52"> - 3" 27'", or 2" 26'". The shortest duration is at the equator, when the sun moves through 18° in P 12"; at the poles it continues several months. When the lat. (of the same name with the deck) exceeds the conipl. of 18° + deck, the sim is less than 18° below the horizon at midnight, or twilight lasts all night, as for ex. with lat. 58° N., deck 21° N. VI. Altitudes. 1. Correction of the Observed Altitudes. 644. The corrections necessary to reduce an altitude observed from the sea-horizon with a sextant or circle to the true altitude, consists of the Index Correction, the Dip, the Correction of Altitude (or the joint effect of refraction and parallax. No. 438,) and, in certain cases, the Semidiameter. SURORDINATF, COMPUSA TIONS. 231 Wlien one of the instruments, No. 522 or 623 is used, the ]3ip is Oiuitterl ; the constant correction sliould be applied tlie fii-st thing. G4.5. The apparent alt. is deduced from tlie observed alt. by ap- plying all the above corrections except rerhiction and parallax. (i4(>. AMr'ii the altitude is less than 1U°, the mean refraclion in Table 31 may be in error more than 1', and should be corrected by Tables ;J2 and 33 if a barometer and thermometer are at hand. For precision, this is necessary in all cases. [ I . ] To Correct the Sun 's Altilmle. 647. At Sea. Ai)ply the Ind. Corr. ; subtract the dip corre- sponding to the heiglit of the eye, I'able 30; subtract the reliaction for tiiis alt., Table 31, to the nearest minute. When the iaicer limb is observed, add 16' to this reduced alt.; when the uppc7- limb is observed, subtract IG'; the result is the true or correcteil alt. of the sun's centre. Ex. 1. Obs. alt. of Q 2S° 54', inJ. Ex. 2. Obs. alt. of 42° 11'. ind. corr. +f, height of the eye i6 feet: re- corr. -17', lu■i^llt of the eye 30 feet: re- quired True Alt. of the centre. quired True Ah. of the centre. Obs. Alt. 28° 54' Obs. Alt. 42° 11' IdU. Corr. + 3 Ind. Corr. — 17 28 57 4' 5+ Dip -4 Dip -5 28 53 41 49 Refr. (for 29°) — 2 Rcfr. (for 42') 28 5. 4. 48 Semid. (lotv. I.) +16 Semid. (upper !.) -16 True Alt. 29 7 TrukAlt. 4. 32 Kx. 3. Obs. alt. Q 10° 4', iud. corr. t 2 , height of eye 18 feet : required the True Alt. of .Sun's centre. Tr.hAlt. .o",3 Ex. 4. Obs. alt. 42° 11', ind. corr. - -17', height of eye 30 feet : required the Tree Alt. of the centre. Tkuf. Alt. 41° 32'. 648. In the open sea, where an error of 2' or 3' of lat., and a corresponding error of long., are of no great consequence, the corr. of ait. for the sun (when liie loirrr liml) is observed), may be taken from Table 38, in which it is •,iveii to tlie nearest minute. X. 1. (Ex. 1 above.) Ex. 2. (Ex. 3 above.) Obs. Alt. Q 28 54 Obs. Alt. Q .0° 4 I:iJ. Corr. + % Ind. Corr. 28 57 7^ 6 Ht. 16 {., Alt. zy" Corr. lit. 18 f., .\1(. 10 \ Con • + 7 Trub Alt. 29 8 True Alt. 10 13 If the upper limb has been observed, proceed as above, ana deduct 32'. Ei. Obs. Alt. 88° 40', Ht. of Eye 30 f., Ind. Corr. - 5 , True Alt. 88° 14'. 649. Accnrateh/. Apply the iud. corr. and (at sea) the dip; rorrect the refr. bj' Tables 32, 33 ; take tlie scmid. and parallax from tile Nautical Almanac; and subtract the piuallax in alt., Table 34.. Miinite acci:racy in ait. at sea euii rarely be worth ti)c tri.(ubl«» 232 NAUTICATi ASTkoNOMt. oestowed upon it, from the uncertain state of the sea-horizon. The examples, No. 651, will serve, supplying the dip. 650. SVhen the altitude of eitlier limb of tlie sun is observed, and the altitude of tlie other limb (which will appear the same in the instrument) is observed from the opposite point of the horizon (No. 635), take half the dirt", of these angles and add to it the correction of alt. ; the sum is the true zen. dist. Ex. 1. Obs. Alt. Q S. 63''49'io", N. 115° 46' zo": required the true Zenith Distance. QN. 1 1 5° 46' 63 49 True Z. Dist. 25 58 59 N. Ex. 2. Obs. Alt. Q N. 81° 59' o", ll S. 97° 40' 30": required the trae Zenith distance. Q S. 97° 40' 30" N. 8. 59 o 2) 154130 Lpp. Zen. Uist. 7 50 4j RelV. + 8 Tkue Z. Dist. 7 50 53 S. 651. On Shore. When the alt. is observed from the quicksilver, apply the ind. corr. at once; halve the result, and proceed as in ^Jo. 649, omitting the dip. Ex.1. Jan. I Ft 1878, alt. (^ir the <i\ irk- iilver .7° 24' 0', ind. corr. - 4 ' so'. lar. 306 inch. therm. M-": find the True Alt. Obs Alt. Q 7° 24' o' Ind. Corr. -4 SO M. Refr . 6' 6 ») Therm. Bar. + 5 7 8 39 35 6 18 r Corr. of Alt -6 9 Par. — 9 Corr. of Alt. 6 9 » T! 26 Semid. + 16 18 Tru eAlt. 8 49 44 Ex.2. July ist, 1878, alt. 060° ir4o'. ind. corr. +2' 35', bar. 29-2, therm. 76°! find the True Alt. Obs. Alt. 60° ii' 40' Ind. Corr. -i. 2 35 M. Refr. j' A'\ 2)! Therm. - 30 7 7 Corr. of Alt. - 1 26 .Semid. ~ < 5 46 RUE Ai.T. 20 49 <;c Ex. 3. May 3d, 1878, obs. alt. in the quicksilver 116° 14' o', ind. corr. + 2' o', bar. 292, therm. 58^ : required the True Altitude. True Alt. 58° 23' 23', Ex. 4. July 9th, 1 878. obs. alt. Q in tlie c bar. 298, therm. 62" : required the True Altitude. ' 17' 50', ind. cor. +54', True Alt. 60° 24' 39' [2. J To Correct a Star or a Planet's Allitudt. fi52. At Sea. Apply the index corr.; subtract the dip and refraction. Ex. 1. Obs. alt of a star 10° 28', ind. r. +2', heis;ht of eye 16 feet: required ! True Alt. Dip 4 ind Refr. 5' True Alt. Ex. 2. Obs. alt. of a star 46° 12', ioA corr. —3', height of eye 16 feet: reqniti>i the True Alt. 46° 12 Sub. 3', 4', and i' 8 Teuk All 46 4 having corrected for iiidei. error, subtract the corr. in Table 38. BlIBORDINATE COMPUTATIONS. 2.",3 Ej. S. Obs. alt. of the planet Venus 30^ 14', ind. eorr. + 3', height cf eye 12 feet : reqaire.1 the True Alt. Jbs. Alt. 30° 1+' Ind. Cjrr. + 3' 1 Table 38, -5 ) ^ Tbce Alt. 30 12 Ei. 4. Obs. alt. of the planet Mtn 78° 57', ind. corr. + 7, height of eye jo feeti required the True Alt. Obs. Alt. 78° 57 Ind. Corr. + 7' 1 , Table 38, -5 ) "*• ^ True Alt. 78 59 (i.').3. Accitrately. Proceed as for the sun, No. 640, omitting seinidianietcr. A star's corr. of alt. is the refraction alone, No. 438, p. 147. For a ])hinet, find the hor. par. in the Nautical Almanac; find the j)ar. in alt. corresponding, in Table 45, and deduct it from the refraction. Ei. 1 . Obs. Alt. of Sinus in the quick- liJver 37' 9' 35", ind. corr. —7' 30", bar. 30-2, therm. 42'' : required the True Alt. :^( Obs. Alt. 37' 9' 35" Ind. Corr. —7 30 2) 17^~~5 iS 31 2 M. Refr. 2' 53") Therm. + 3 Corr. —2 57 True Alt -^ 57 Ex. 2. Obs alt. of <t Polaris in the mercury 102" 38' 30", ind. corr. + 1' 30", therm. 62^^, bar. 30 inch. * Obs. Alt. 102° 3«' 30" Ind. Corr. + J 30 M. Refr. o' 46"-8 ) Therm. — ' "^ 1 Corr. o 45 -6 / _ i) 102 40 o 51 20 o 3. Dec. 2ist, tS-S, obs. alt. Venus fllh, 1878, obs. alt Ma the 16° 48' 40', ind. corr. j in the quicksilver, 41° 49 30 + r 40', bar. 298, therm. 62°: required the True Alt. Venus' H.P., p. 277, N.A. 5"-2 Obs. Alt. 116" 48' 40' Ind. Corr. + i 40 0..6 ■<; M. Refr. o' 3 5''9\ Therm. —0-9 I liar. -0-2 o 34 sr Far. -2-6 Corr. of Alt. o 32 -2/ True Alt. 58 24 38 [3.] To Correct the Muon's Alliiude. + I ■ 20", bar. 29-2, therm. 58 the True Alt. : requ Mars' H.P., p. 278 N.A 5'' 5 Obs. Alt. Ind. Corr. 4'" 49' 3'^' + 1 20 2141 50 50 M. Refr. 2' 3.'Sv 20 55 H Tii.-rm. - 3 liar. - 4 Par. ^-:! Corr. of Alt. 2 19 •7'' True Alt. 20 51 5 6.54. At Sea. Find the (Irecn. l):ite roughly, and take out of the Nautical Almanac the hor. ))ar. and seiuid. to the nearest noon or midnight. Aj)ply the ind. corr. to the alt., subtract the dip; when tlie lower limb is observed, add the semid.; when the upper limb is observed, suhtinct it ; the result is the iipp. alt. of the centre. With the A. alt. and hor. par. find, in Table .'50, the moon's corr. of alt., which add. The result is the true or corrected alt. of tlie moon's centre, aj)j)roxiuialely. 231 KAtJTICAT. ASTRONOJTT. E^;. 1.* Mayisth, 187S, long. 5.-= W., at 8'' 4*'" !■•"•. obs. alt., ^ 37" 10, ind. ciirr. + 3', height of eye 14 feet: retiuired •he True Alt. Es. 2. Sept. i8th, 187''. at jh A.M., obs alt. J 6." eve 16 feet, ind. corr. —3': Alt. 20', height 0} find the Tnw ThcGi-. Date is 13th, 11" i o"', H.P. at The Gr. Date 17th, 3!" 20'", H.P. M midnight 60', semid. 16'. Ind. Corr. + 3' ) 37° '°' noon, 55. semid. 15'. Ind. Corr. - 3') 61" 20' Dip - 4 Semid. + 16 ) 37-25'. H. P. 60' True Alt. + >5 37 ^5 38 11 Dip - 4 Semid -.5) «,°c', H.P. 55' True Alt. 60 s8 + 26 6. Z4 Ex. 3. Jan. 3d, 1878, long Ind. corr. + 3' 159° E., at 9" 10"- P.M !) 85° 42', height of eye 20 feet, True Alt. 86^ 1'. Ex. 4. July 5th, iS78,long 18 feet. 172° W., at 3' A.y..: J 14" 28', ind. corr. 0', bright of eye True Alt. 15" i'. 655. Acr.urateh/. (1.) Reduce the hor. par. to the Gr. Date, and find the semid. Table 40. Reduce the par. by Table 41, and augment the semid. Table 42. (2.) Take out the refraction for the limb observed, correct it for baroni. and therm. ; subtract this corrected refraction from the alt. and apply the augmented aemidiameter. (3.) To the log. sec. of the alt. thus reduced add the prop. log. of the reduced hor. parallax; the sum is the prop. log. of the parallax in alt. This par. added to the reduced alt. gives the true alt. of the centre. As, however, the degree of precision obtained by these prece])tj will rarely be required, we shall, in the following examjile, employ Table 39. Ex. 1. July 30th, 1S78. lat. 42° S., long. 42° 13' W., at 5'' 36 ' 39' 50", iud. corr. +2' 17", height of eye 22 feetj therm. 72 True Alt. e Gr. Date, 30th, S^ I r 30- H.P. 30th, Noon 59 55"6 30th. Midn. 60 6 2 I2-I.uurly Var. + 10 -6 8" 14", var. io"-6 + 7 •» 59 55 6 Eq.iat. H.r. 60 .8 Red. for Lat. "5 '2 Ued. U P. 59 5' f I corresp. to 59' 58 ' I 5' 21 Augment. 10 Aug. Semiil. 16 31' Obs. Alt. Ind. Corr. + 2' 17 Dip. -4 30 Aug. Semid. 36° 50' and 59' 4 58" iprin. 72°, sub. 3" M.T. obs. alt. T) :gi : required tha 36 37 37 + ■63 ' 36 54 8 45 56" — 2 _+4L 46 41 46 46 4 46 46 37 40 54 656. When the moon i.-; referred to the opposite point of the horizon. No. 535, half the ditf. of the alt. and its supplement is the zenith distance of the illumin..ted limb, to which ti:e augmented * The examf les being given merely iu illuatration of the nilfcs, no regard huH been p~id to the vuibility cf the 0100a at the time and place specified. SrnORDINATE COMI'UTATIONS. 2 "5 seinid. fa to be apjilieil the contrary way to that direoicil for tlie alt. In certain cases botli limbs can thus be observed, No. 540, and the semidiametcr avoided. "2. To Reduce the True to the Apparent Altitude. (1.1 For the Sun, a Star, or a Planet. 657. Take out the refraction to tlie true alt. as if for the aoj). alt., correcting; it, when necessary, for the baroin. and therm. ; sub- tract the parallax in alt., add the remainder to the true alt., and subtract the correction in Table 43. [2.] For the Moon. 6.58. Find her corr. of alt. for the true alt., as if for the app. alt., and apply the corr., Table 44. Ex. J 's Hor. Par. 59', True Alt. 48° 41' 12" 48° 41', and so', — -iS' 6") „ Corr! Tabic 44, i 28 ) " ^^ 34 App. Alt. 48 2 38 fi.')?). To reduce the app. alt. to the observed alt. for a particular instrument and given height of the eye, apply the ind. corr. the vjiposile way, and add the dip. 3. Reduction of Two Altitudes to an Intermediate Point of Time. ()60. Two altitudes observed at periods of time not distant, at^brd, by simple p.-oportion, the altitude at an intermediate time. (1.) Find the interval between the time of the 1st alt. and the time proposed, and call it the partial interval. (2.) To the prop. log. of the partial interval add the ar. co. pro]), log. of the whole interval, and the prop. log. of the ditf. of alts.; the sum is the prop. log. of the change of alt. in the jjartial interval. (3.) When the 1st alt. is the lesser, add this change; when it is the greater, subtract the change. Ex. 1. At lo*" iS™ 4" by watcb, obs. an alt. 54° 56'; at 10*1 29"' 1 1» obs. a seconJ ilt. 55" 12'; reijuircd the Alt. at 10'' 23"' 6». Alt. 54^56' "-.0^.8^ 4- j 3,,,, ,,.,„,. ,„3 55 '^ 10 29 II ) II 7 ar. CO. i>. log. 8-791 Diff. 16 I"-, log. 105' Change of Alt. 7' (T. log. 1-395 Alt. req. 55 3 Ex. 2. At 12'" s?"" 24* by watch, obs. an nit. 39° 2'; and at i'' 8"" 18' obs. a second alt 36' 42 : required the .Ml. at i'' i"' 29*. Change of Alt. —0° 53', and Alt. req. 38° 9'. Ex. 3. At c'' ;8'" 36' by watch, obs. an alt. 47° 33', and at i*' 5" 47*, obs. a second alt. 4.7' 52' I required the Alt. at i'' i'° 29*. Change of .\lt. + 8', and .\lt. req. 47° 41'. The altitu<ie thus deduced differs from the true alt. by a propor- tional purl of tlic 2d dilf. of alt. upon tlic interval, No. f'OS. The 236 NAUTICAL ASTKONOMT. metliod serves very well wlieu the azimuth is large, or the object 60° or more from the meridian, or less if the interval be small ; but in cases near the meridian the result will be sensibly in error, unless the interval is very small. The error arising from the neglect of tlie '2(1 difF. will be less as the intermediate time is nearer to the beginu ng or end of the interval. 4. Reduction of an Altitude to another Place of Observation. 661. The run of the shi]) in tlie interval between the taking of tlie two altitudes which constitute certain observations, renders it necessary te reduce one to the place of the other. When the ship approaclies the sun directly she raises him 1 for each mile of distance made good. When the sun bears obliquely (as for ex. 3 points) from the course made good, if we consider the angle between this last course and the sun's bearing (or 3 points) as a course, the space by which the ship approaches tiie sun is the D. Lat. corresponding to her Dist. made good.* When the sun"s bearing is at right angles to the course made good, the ship neither approaches nor recedes from him ; when the bearing is abaft tliis line, she drops the sun. When it is j-equired to reduce an alt. observed at 1 o'clock (for ex.) to what it would have been if observed at the place where the ship is at 2 o'clock, the ship having approached the sun, we have merely to add to the alt. observed at 1 o'clock the jiortion of sjiace or arc by which the sliip would have raised the sun in 1'', if lie had ))reserved his bearing at 1 o'clock unaltered. Hence the following rules. To reduce the 1st alt. to the second place of observation. (1.) Take the diff. between the bearing of the body at the first observation and the ship's course, as a Course, and the dist. run as a Distance; the D. Lat. corresponding is the reduction for run. (2.) When this course is less than 90° or 8 points, add the red. to the first alt. ; when the said course exceeds 90° or 8 points, sub- tract the red. ; the result is the alt. reduced to what it would have been if observed at the second ]ilace of the spectator. If the ship does not preserve the same course, the course made good must be employed. As it is difference only of bearing or azimutli that enters into this question, the vai-iation (supposed the same at both observa- tions) is rot considered; but if the ship's course changes, the deviation should be atttnded to. Ex. 1. Observed the sun's alt., the sun bearnif; S.E. by E. } K. the course E. by N. A N. (by compass). Sailed for i' 15'" at the rale of 7^, knots : retjuired the Uedueti.in of the Alt. for Run. From S.E. by E. \ E. to E. i< 2{ pts. ; from E. to E. by N. \ N. is \\ pts. The course 4} points, and dist. 94, give D. Lat. 6'3 the Reduction to be added to tlic Alt, • As the distance is des:'ribed upon a spherical surface, in stiictness a correction is neces- sary ; also the dist. made [.'ood on the spiral rhumb should be reduced to that on « ;;re;i« circle ; hut these rcHncmeuts aie generally iucou,:i;teut with the x-Mcdata of the qu««ti'T»^ BViHORDlNATF. COMPUTATIONS. T.)t Ex. 2. Sun South, alt. 55° 3o''5, course E. by N., rate 6-8 knots, interval 12" : rcilui« the \U. for the Uuii. The su|)|)l. of 9 pts., or 7 pts., and (list. 1-4, give D. Lat. 0-27, or o'-j, which subtracted from 55° 3o'"5i gives 55° jo'-z, the Alt. required. Ei. 3. Obs. sun's alt, sun bearing N.E. J E., course N.W. ( N , sailed for 36'" io« at the rate >f io'2 knots: required the Reduction for Run. The Rkduction is o'o. Ex. 4. Obs. a star's alt. 37° iS 40", bearing S.E. by E. 4 E., course N.W. by W. 4 W., rak 58 knots, internal 2'' z^"' : reduce the Alt. for Run. The Reduction is I'i'g to tub. ; the Am. 37°4''8. When the course at the 1st observation is directly towards tlie sun, the (list, run in the interval is tiie correction, and is to be added to the 1st alt. ; when directli/ from the sun, to be subtracted. Kx. Obs. sun's alt. 29° 7' 30", bearing E.S.E., course E.S.E., rate 5'4 knots, interval 3'' 6'" : reduce the Alt. for Run. The Reduction is i6''7 to add; the Alt. 29° 24'"z. fi62. To reduce the 2d alt. to the first place of observation. Take the bearing at the last observation ; find the reduction of th(! alt. as above, and ap])ly it to tlie 2d alt. the contrary way to that tlirectcd in (2) above. Ex. 1. Observed the sun's alt., sailed S.S.W. for 48'" at the rate of 34 knots, when the 2d alt. was taken, the sun bearing W.S.W. : required the Correction of the Alt. for Run. From S.S.W. to W.S.W. is 4 pts. The course 4 pts., and Dist. 2-8, give the D. Lat. 2*o to be subtracted from the 2d Alt. Ex. 2. Course N.W. by N., obser^-ed the sun's alt. After sailing for i'' 36" at 82 knots, observed the 2d alt. 39^44', the sun bearing E.S.E. From N.W. by N. to E.S.E. is 13 pts.; then the course 3 pts., and Dist. 13-1, giv» D. Lat. 10-9, which added to 39° 44' gives 39° 54''9, the Alt. reduced. When tlie course at the 2d observation is directly towanls the sun, tlie dist. run is tiie correction, and is to be subtracted from tho second alt. ; when directly from the sun, it is to be added. 5. To find the Altitude. [1.] Oh t/te Meridian. 6G.'3. For the sun, the moon, or a planet, find the time of Mer. Pa.ss., No. 623, &c., and reduce the declin.. No. 579, &c. Find the rohit. \\'hen the lat. and decl. are of the same name take the sum of the colat. and decl. ; when of different nam(!S, their ditf. ; the residt is tiie mer. alt. If the sum exceeds 90° take its complement. IJclow the Pole. Find the pi)l. dist., and subtract it from the latitude. [2.] On the Prime Vertical. 6f)4. By Inspection. See Table 29, and Explan. of Table 5. 66.'). Jiy Computation. (1 ) Find the approx. time of Passage, No. 6:j(J; to this reduce the declin., in the case of the sun, moon, or a jilanet (2.) Add together the log. sine ot the declin., and the log. cosec. 'A'iht lat ; the sum is tin; log. sine of the true alt. required. 238 NAUTTCAL ASTRONOMY Es. 1. Ju'y 12U1, 1S7S!, l,it. 5i''4S'N., king. 4° 56' W.: find tlie Suns Alt. on the Table 29, Lat. 51°, Decl. 21°, ,. ,... Hour-angle, or App. Time/ ^^ *" ■ Long. 4'- 56' W- + 20 Green. Date nth, 5 6 © Decl. 1 2th, zi-'ss'N .3th, 21 50 N Daily Var. S Daily Var. S' and 5'- gi^ es 2', whence Red. D.-cI. is2i° 56' Deol. 21° 56' sine 9-57I52 Lat. 51 48 cosec. o-rov66 Alt. 28 23 sine 9-676^8 Decl. 38° 40' sine 979573 Lat. 50 48 cosec CI1073 Alt. 53 44 sine 9-90646 Ex 3. Lni. 46° 14' N. : find the Alt ol Capella on the Prime Vertical. Decl. 45° 52' sine 9'85596 Lat. 46 14 cosec. 0-14136 Alt. 83 38 sine 9'997 3i [3.] To find the Allitude, the Hour-angle being fhen. 666. By Inspection. See Explan. of Table 5. 667. JBy Computation, Having (in tlie case of the sun, moon, or planet) foiiiitl the Or. Date and the declination. Take tiie sn]ipl. of the hour-angle to 12''; add together the pol. dist. and colat. Add together the log. sine square of the suppl. of the hour-angle, and the log. sines of the pol. dist. and colat. ; the sum (rejecting tens) is the log. sine square of an auxiliary arc x Write X under the sum of tlie pol. dist. and colat. and take the sum and diff, and half tlie sum and half the ditf. Add together the log. sines of the last two terms; the sum ^rejecting tens) is tlie log. sine square of the zen. dist. Ex. 1. Lat. 2 2° .5'N.. decl. 2° 49' S., hour-angle 2" 14"' 3 (working to the ne arest minute). Hour. angle ,1, .^n, 36. Suppl 9 45 z+ sin. sq. 9-96200 P. Dist. 92 49' sine 9'99947 Colat. 67 45 sine 9-96639 Sun, 160 34 Arcx •33 57 sin. sq. 9-92786 Sum -94 3' Ditf. 26 37 4S. 147 15 sine 9'733'* iD. 13 .8 sine 9-36'S2 Zen Dist. 41 ' '9' sin. sq. 9 09500 Ei. 2. Lat. 35° 15' N., decl. 20° o' N., hour-angle 4'' 53'" 19*. Alt. 24° 41'. Ex. 3. Lat. 19° 20' S., decl. 19° 20' S., hour-angle i^ 18'" lo'. Alt. 71° 35'. When the lat. is 0, we may use either N. or S. pol. dist. When the declin. is 0, the pol. dist. is 90°. When both lat. and declin. are 0. the z. d. is the hour-angle converted into arc. Ex. 1. Lat. o, decl. 23° 27' N., hour-angle 4"^ 30"' 14'. Alt Ei 3 Lat. 30' o N., decl. o, hour-angle 3'' 38'" 3c' A;.T. 30"*-. SUBORDINATE COM PTITATIONS. 239 [4.] nfnii the Altitude, thcAiimnth being gteen. 668. Add togctliei- tlic log. sine of tlie aziiii., tho log. cosine of Uic lat., and tliu log. sec. of the decl. ; the sum (rejecting tens) is the log. sine of an angle A (see note to No. 616), p. 222. Under A luit tiie azini. reckoned from the elevated pole; tako half the sum and half the ditf. Take half the sum of the pol. dist. and culat. Add togetlier the log. tan. of this half sum, the log. cos. of the lialfsum of th, azim. and A, and the log. sec. of tiieir half diti". ; tiie sum (rejecting tens) is the log. tan. of iialf the zen. dist. Ex. Lat. 51° 30' N., decl. 20° z' N., azimuth S. 69° 39' W., that is N. iio^ 21' W.i rrquireil the Alt. Az. 69° 39' sin. 9-97201 Colat. 38° 30 Lat. 51 30 COS. 9-794'5 P. Dist. 69 58 Decl Su a lug 28, i S. 54° 14 tan. 0-14246 .\ °, .r ^^' sin. 9-793^7 74 J3 COS. 9-43007 Az. 35 58 sec. 0C9186 Sum i+s'tI, JS. 74^ ^3' ^"47' tan 966439 Ditr. 71 56. J D. 35 58 For other Exa Zen. Dist. 49 34 Alt. 40 26 mples reverse those in No. 674. 6 To find the Change of Altitude in a Small Interval of Time. [ 1 . ] The Hour-angle and Altitude being given. 669. (1.) When the body is to the E. of tiie meridian, swJ^rarr half the interval from the hour-angle; when to the W. of the meridian, add half the interval: call the result the reduced hour-angle. (2.) Add together the log. cosines of the lat. and declin., the log. sine of the red. hour-angle, the log. sec. of the alt. and the log. sine of the interval ; the sum (rejecting tens) is the log. sine of the change of alt.* (3.) When the body is to the E. of the meridian, add this change to the alt.; when to the W., subtract it: the result is the alt. required. Ex. 1. Lat. 51° 30', decl. 22° 20', trae Ex. 2. Lat. 5 1° 30', decl. 22° 20', tnw alt 44° 47' 36", hour-angle 3'' 0'" o» to the E of the meridian: required the Alt. 10" alt W 44^ 47' 36", hour-angle 31' 0"' o* to the : find the Alt. 2o"' afterwards. »l"b,rwards. Hour-angle 3'' Co' E. lat. cos. 9-7942 3" o'"o'W. lat. COS. 9-7941 Half-int. - 5 decl. cos. 9-9661 A -I- 10 decl. cos. 9-9661 Red.ll.ang. 2 55 sine 9-8398 3 10 sine 9-8676 44" 47' 36" sec. 0-1490 int. sin. 8 6397 Chan-gk 1 24 I sin. 8-3888 44" 47' %(> 2 59 20 sec. 0-1490 int. sine 89403 sin. 8-7172 Alt. 46 11 37 LT. 41 48 16 The true alt. U 46° 12' 48", or the The true alt. i 41° 52 24", or the error procsii is here 1' 1 1" in defect. is 4' 8 ' in consequence of the length of the interval. • The prop. logs, may be used for the sines of the small arc and the interval, provided that the arithmetical complements of all the other quantities be employed, and the const, s -8239 added. The proper logarithm for the jiurpose is the log. of tiie small arc or the interval in seconds of arc ("). The inaccuracy attending the use of the sine, instead of iti ■re, in these computations is insensible, as the sine of 1 falls short of its arc by oidy p"*a^ the roe uf 2 by 1 "'5, aud that of 3 by 2"-9, or o*iy of tune. *240 NAUTICAIi ASTRONOMY. The method is more accurate as the object is more nearly E. or W. Thf proper alt. to employ in this computation is the middle alt. between those at the beginnins: and end of the interval; for greater accuracy, therefore, the work should be repeated with a new alt. tlius deduced. [2.] The Azimuth being given. 670. By Inspection. Multiply the change of alt. in 1"" of time, Table 46, by the interval, botii being in min. and decimals. Ex. Lat. Si", azim. 72°: find the change in Alt. in 3™ 12*. The change of alt. in i" is about 8'"7, which multiplied by 3-2 gives 28', the CaANoa required. 671. By Computation. Add together the log. sineof the azimutli (reckoned either from N. or S.), the log. cos. of the lat., and the log. sine of the interval of time; the sum (rejecting tens) is the log. sine of the change of altitude. It is more correct to use the azinmth corresponding to the middle of the interval of time.* Ei. Lat. 5 1° 49', azimuth of Arcturus 7 in 2-" 51'. 1° : find the change of Alt. in s"- 12', and also Az. iz" sine 99782 Lat. 51 49" COS. 9-7911 Int. 3"' IX' sine 8-1450 Change req. 28' 13" sine 7-9143 9-9782 9-791 1 Int. 2™ 5.« 80946 Change req. 25' 8" sine 7'5639 672. All bodies on the same or ojjposite azimuths change their altitudes at the same rate, whatever be their declinations. VII. Azimuths. I. To find the Azimuth, the Altitude being given. 673. By Inspection. See Explanation of Table 5. 674. By Computation. Add together the pol. dist., the lat., and the alt., take half th3 sum.f and take the ditf. between this half sum and the pol. dist. Add togetlier the log. sec. of the lat., the log. sec. of the alt., the log. cosines of the half sum and remainder; the sum (rejecting tens) is the log. sine square of the azimuth, J to be reckoned from the S. m N. hit., and from the N. in S. lat. * The above rules, Nos. 669, &c., relate to the change of the true altitude. To compare the change of alt. as shewn by an instrument with the true difference, in a given interval of time, a small correction woulil, in general, be necessary, on account of the change of refracti jii, and in the case of the moon, for tlie change also in her parallax in altitude. t The learner will observe that in this formula the pol. dist., lat., and alt., occur in tlie reverse order of that in No. 614, in which last their initials form the word alp. The 2d and 3d terras take secants ; the last two, cosines. X The angle obtained is the siipjjlement of the sagle P Z A in fig, 1, p. 102 SU BOH DrN ATK. COM P uTATIOMS. V.\. 1. Ijit. 5i°3o'N., alt. 4o°2?'to\iie W , *!t*-rj. 20*2' N. : required the Aziiiiutli. I'ol. Oisl. 69" 5S' l^t. 51 30 sec. 0-10585 V'L .^o 25 800.0-11842 >(" 53 80 56 J COS. 9-19711 10 58 j COS. 9-99198 AZIMUTH, S. 69° 39' W. sin. sq. 9-51336 Wlien tlie lat. is 0, if ti.e (Jecliii. is N. rt'fkoiieil from llie soiitli ; it' it is S. froin the iiurlli. W'licM the decliii. is 0, tlie aziiiiutii is •'eckoiiL'<i from l&t., and fi-oiii the S. in N. lat. Ut. 40° 8' S.. lVi'i. 1 1-0' N.. til. •Xstward .- requbed the At m. IMCTH, N he N. in S. Es. I. I>it. 0°, decl. 23° 27' Ex. -2. Lat. ii°i2'N„ decl. When botii tlie lat vertical. ;., alt. 4i°2' W. AziM. N. 12 1° 50' W., or S. 58" 10' VV. 3=, alt. 54° 30', to the East. Azim. S. 73° 53' E. nd dtnd. are 0, tlie ohjcct moves on tlie |)riine 2. To find the Azimuth, the Hour-angle being given. 67.5. (1.) Take half the sum of the pel. dist. and colat., and half 'he difference. (2.) Add together the log. cot. of half the honr-angle, the log. sec. of the half snni, and log. cos. of the halfdiff.: the sinn (rejecting tens) is the log. tan. of half the sum of the azimuth and another angle A. \VlR'n the half sum of the pol. dist. and colat. exceeds 90**, take the su]j])l. of the resulting arc for the half sum rerinired. To the log. cot. already employed add the log. cosec. of the half num. and the log. sine of the half diff. ; the sum (rejecting tens) ig the log. tan. of half the diff. of the same two angles. (3.) The sum of the resulting half sum and halfdiff. is i\\e greatei of the said two angles ; the difference is the lesser. When the ]>ol. dist. exceeds the colat. the greater of the two angles is the azimuth rerpiirod ; when the pol. dist. is less tiiaii llie colat., the lesser of the angles is the azimuth rerpiired Ex. I. Lat. lo'io'N., decl. 22° 14' .S., hour-angle i'' 44'" I7"\V. : rei,uiieJ the Azi'niuth. H. Angle i ''44"'i:" cot. 3-63548 cosec. 0-00235 n tan. o-o8i59 91 26 (suppl.) 5° 37 Sum N. 142 3 W. Azimuth (\t. dist. exceeds col.) Diir. 40 49 the other Angle, or A. ' S., decl. 1 1" 18' S., hour-angle 5I' 11'" 20" : the Azimuth 91" b', »!»« Half 52 8 cot. 0-63548 I' Dist. 112" 14' Colat. 79 40 Sura 191 54 Diir. 3» 34 J.S. 95 57 sec. 0-98439 4D. ■ 6 ,7 cos. 0-9S222 ii^ ii' tan. i-6u2Q9 El. 3. Lat. 13" 52' N., decl. 46° S' N., hour angle 1 1- E. of Me ifit'k B. 242 NAUTICAL ASTRONOMT. 3. To find the Azimuth, the Hour-angle and Altitude heimj giiJen. 676. Atld together the log. sine of the pol. dist. (or log. cos. of the decliii.), the log. sine of the hour-angle, and the log. sec. of the alt. ; the sum rejecting tens is the log. sine of the azimuth. Ex. 1. Hour-angle i 58° 40', pol. dist. 104." 24 Aamath. .9- .9-, alt. : required the Ex. 2. Hour-angle o'' 46'" 39% alt. 6fo', decl. 14° 24' (N. or S.) : required tlie Azimuth. Pol. Dist. sin. Hour-angle sine Alt. sec. AziM. 39°ii- si 9-9S6. 9-5305 0-2840 1. 98006 Decl. COS. 9-9861 Hour-angle sin. 9-3057 Alt. sec. 0-3430 AziM. 25^ 33' sin. 9-6348 Tills nictliod can lot sliew wlie tlier the body is to the N. or S. of 4. To find the Azimuth, not far from the Meridian, hj the ohseived change of Attitude in a small Intei-vnl of Time. Qll. liij Inspection. Divide tlie given chanoe of alt. by tlie interval, in niiii. and decimals; the quolient is the cliaiige of alt. in I™. With tliis change and the lat. enter Table 46, and take out tlie aziniiilli, which corresponds ajiproximatcly to the middle of the interval. Es. Lat. 35° ; the change of alt. in 20°" 12' is 59' : find the Azimuth. 59 divided by 20-2 gives 2-9, the change of alt. in i"', which gives tlie Azim. about 14°. 678. By Computation. Add together tlie log. sine of the change of alt., the log. cosec. of the interval, and the log. sec. of the lat. ; tl'.e sum Is the log. sine of the azimuth about the middle of the interval. Ex. \. Lat. 51° 26'; in s" 20* observed Z2' change of alt. : required the Azimuth. D. Alt. 22' sine 7-8061 Int. 5"' 2o' cosec. 1-6332 i/at. 5i°26' sec. c-20<;2 Azim. 26° 10' sine 96445 At about 3"' after the 1st observation. Ex. 2. Lat. 34 40'; in 20"' 12' observed 59' 6" cliange of alt. : Teejuired the Azimuth. D. Alt 59' 6" sine 8-2353 Int. 20"' 12" cpsec. 1-0554 Lat. 34° 40' sec. 0-0849 Azim. 13° 44' sine 9-3756 At about 10"' after the 1st observation. 679. This method will sometimes be useful, as for determining the variation, but it must be employed with caution; the interval should not be very small, the body- should not be far from the meri- dian, and both alts, must of course be observed on the same side. The degree of dci>endaiice is easily estimated by changir:g the diff. of alts, by the amount of probable error, as about \' or 2': Thus, 1' error of diflf. alts, produces in Ex. 1 an error of \°\, while in Ex. 2 it produces an irror of only 14'. * * The work of finding the Azimuth is much lessened by the use of suitable tablo.-s. Burdwood and Davis's Azimuth tables and Star Azimuth tables extend from the equator to 60" latitude, aud are published in a convenient form by J. D. Potter, H5 Minones, London, E. Such tables are itdispcnsable for the navigalian of iroushiis. i'tt also Lecky's " Wrickleis," for stars. CHAPTER V. Finding the Latitude. 1 Hf TUB MuKiuiAN ALTiTUDt;. II. By the IIeduction to thb Meridian. III. By Douule Altitude op the same Body. IV. By Double Altitude of different Bodies. V. By thh Altitude of the Pole Star.* 080. The pole lemains always in the same absolute fixed position from whatever point of the earth's surface it is viewed ; its altitude at any particular place is, therefore, always the sauie. The position of the equator, which is 90° from the pole, is also always the same at the same place, and is determined by reference to the celestial bodies, whose declinations are measured from it. The latitude of the place niay.tiierefore, be determined directly by observation, and indejiend- fiilly of the latitude of any other place. When the body observed is on the meridian (at which time its dititude ceases to change) the time is not noted; but if it is not on the meridian, either the absolute time must be given, or a second BJtitude must be obtained after a measured interval. I. By the Meridian Altitude. mplest, and in general the most satis! the latitude, is by observation of tli celestial body when on the meridian of the place. 'I- fi8I. Tlie simplest, and in general the most satisfactory, method of deterwiining the latitude, is by observation of the altitude of ii ridii * The several methods of latitude which are given in this work under the heads enuriic- ratod above, and which may be considered as distinct methods, of which the solul'cm deiieiuli an ciriunistances as elsewhere described, amount to eiRht. The seaman, who will reinenili<r the .idiii;!", " lead, latitude, and look-out," scarcely iimls tn lie- n-iiiiiidid thiif IIm- hitilmlc is oltrn the only clement necessary, — that headlands en v,i-i ir;i. is ..t . mi-i n. ;i|>|.rii:h l.i il, nnn i.iiimrims passages or channels taken, by reference In !, Ir ;il..iic, ;in.| ihii tlh iimc, and •Iririfnrc the longitude itself, depends on the Intitml' . In tin r il,i\^, ,.1", nl.c u -u, Ij -nut and continued velocity is Htta-npfl, in «»r- nn.v.^-'fN', ii :■ i^ol LiciliUrs aic Uiiii.mtK.l lor ilr- tiTiiiining the place oi • . i .|, Tr.- ' : i:,. i-iiin iiccoi'dingly should be furoislievi. nith a method of liiiii : j ..nii lit and satisfactory) adapted ti tverjr occasion that ih .. . ■ i-l : >!i, mi i n i t The manner of dc^i'.i.ni; i,k I.im ,iIl- h..in lui uui. ,di and declin. is fully described Ifl No. 152. 244 NAUTICAL ASTRONOMY. 1 Meridian Altitude of the Sun. 682. The Ohsrrvation. When tlie sun is near the nieri<liaii, con- tinue 10 observe the altitude till it is found to decrease ; xhe ijreatest alt. reached is the nier. a!t.* In latitudes above 66*^4 tlie sun, being above the horizon tlie whole 24 hours during; part of the suiniuer mouths, may often be observed below the pole at midnight; in this case the smallest altitude is the mer. alt.f When accuracy is required, note tlie baroin. and therm. 683. The Computation. At Sea. (1.) Take the sun's dec!, from the Nautical Almanac, page I., or Table 60, for the noon of the day, and reduce it by Table 19 for the longitude by account. (2.) Correct the alt. for index error, dip, semidiameter, and refraction, No. 647 ; subtract it from 90**, the remainder is the zenith distance. (3.) When the observer is to the N. of the siin, call the zon. dist. north ; when he is to the S. of the sun, call it south. When the zen. dist. and decl. are of the same name, take their suvi; when oi contrary names, take their difference: the result is the lat. When the decl. and zen. dist. are of the same name, tiie lat. is also of that name ; when the decl. and zen. dist. are oi different names, the hit. takes the name of the r/reatcr.X Ex. 1. May 3d. 1902, long. 38= W. , Ex. 2. July 4th, 1902, ' obs. Mer. Alt. Q 56° 10' to the soutliward, obs. Mer. Alt. Q 81° 59' bearing north, Ind. corr. +2'. height of eye 20 feet: re- ind. corr. o, height of eye, 16 feet: required quired the Latitude. the Latitude. Decl. 3d, Table 60, i5'-29' N. Corr. for 38^ W. -n Red. Declin. 15 31 N. Obs. Alt. Q 56" 10 ind. Corr. + z' 1 Dip -4 } _Zl App. Alt. 56 8 Refr True Alt. Zen. Oist + .6 j 5" ^3 33 37 33 37 N. 49 8 N. Decl. 4th 22°J7' N. Corr. for 101° E. Red. Declin. iTTs N. Obs. Alt. 81° 59 Table 38, True Alt. 82 II Zen. Dist. 7 49 7 49 S. Latitude i"i 7, N. * At sea it is usual to keep advancing the index till the sun has dipped, but it is better to take separate altitudes. t Since the sun, moon, and planets, change their declinations, the mer. alt. is not always the maximum or miiiimiim altitude. Near the equator the ditference, which is as the tangent of the latitude nearly, is very minute. In lat. 60° the sun's alt. will be ma,\inium, in the extreme case, at half a min. from the meridian, and the altitudes will differ only 0"-4 ; in the •ame latitude these quantities will be, for the moon, 7' and 2' respectively. As 0"-4 is inap- preciable by ordinary instruments, and as the moon can be employed for approximation only, it i« not necessary to tabulate this correction- i A ship, on board which the declination had been applied the wrong way, made the Orkney Islands, in coming from the westward, instead of the Channel. A few years agn a •hip buund homewards from .Australia rouod C. Horn got too far to the southwaid ; a similar FINDING THE LATITUDE. 215 Ex. 4. July IJtIi, 1902, long. 49° W ,s. nicr. alt. (^ 89" 44' N., in<l« err 4', eye 18 fi-et : find the Latitude. Decl. 13th Corr. for long. 49° lUd. D.cl. W 21° 56 N. 2. 55 N. Obs. Alt. ij^ Index + 4'l Table 38 + 12 f True .\lt. S9° 44' + 16 90 Zen. Dist. Decl. 21 55 N. When the declin. is 0, the zen. dist. is the latitude ; and whe the zen. dist. is 0, the declin. is the latitude. Kx;!. Ward. 2Ist, 1902, long. 15° \V obs. iiicr. alt. '■' 48° 16' bearing N., ind error - 5', eye 16 feet . find the Latitud Decl. 2I.st 0° I' S. Corr. for long. 15° W. -: Red. Decl. o o True .' Zen. Di.-t. Decl. Ex. 5. March 21st, 1902, long. 60° E., obs. mer. alt. eye 20 feet ■. reiiuired the Latitude, lied. decl. 0° 5' S. Ex. 6. Aug. .'ith, 1902, long. 47° W., obs. mer. alt. J_ 72° eye 16 feet. Hed. decl. 17° 8' N. True alt. 73° l'. Ex. 7. March 20!h, igo2, long. 90° W , obs. mer. alt. eye 12 feet. Ited. decl. 0° 19' S. 1 rue alt. 90°. Ex. 8. Jan. Isl, 1902, long. 138° W., obs. mer. alt. 89° 55' S., index error +2', eye 12 feet. Red. decl. 23° 3' S. True all. 90° lo'. Lat. 23° 13' S. Ex. 9. June 20ih, 1902, long. 172° W., obs. mer. alt. 52° 18' S., index error -2', eye 60 feet (the top). Red. deel. 23° 27' N. True alt. 52° 23'. Lat. 61° 4' N. Ex. 10. Feb. l8ih, 1902, long. 71° E., obs. alt ©'s centre (by bisecting the cloudy disc, No. 539), 48° 22' S., eje 18 feet. Decl. 11° S5' S. True alt. 48° 17'. Lat. 29° 48' N. Ex. 11. Dec. 20th, 1902, long. 160° E, obs mer. alt. (T 28° iS' S., above the sea horizon 2^ miles distant, eye 20 feet. Red. decl. 23° 25' .S. True alt. 28=' 26'. Lat. 38° 9' N. (i84. When the sun is observed below the jiole (at midnight), instead of subtracting the true alt. from 90", add 90° to it ; the lat. vrill be of the same name as the declin. c.6° 26' N., index erro 2', ■rue alt ■^^ 21'. Lat. .33" 26 S. 72" 47' N., index . 0° 9' N.' 8q '48 S. index r _ I', Lat 0" 19 S. Ex. 1. June 5th. 1902 long. Z9° E. Ex. 2. Nov. 13111, 1902, long. 98' W. at 12'' Ksi, obs. mer. alt. • below t ,e pole at 12'' 1- M , obs. mer. a t. j_ below the pole 3° 38' N., ind. corr. ^ 2',teight of eye 20 s° 37' S., ind. corr. - 2', height of eye 30 f.ei : reijuired the Laiitude tei t. Red. Declin. No. .579 (2), 22" 31' N. D elin. N,>on 17° 48' S, Obs. .Mi. . In<l. Corr. + 2', 3 3S' Curr. far ,2'',„WS', „s \v.,„;,/4f '^ D,p. - 4 1 -2 ~3 36 Ue.l. DeWm. iS .S. Obs. All. 5" 37' lUfr. -I.i'l Ind. forr. -2', Seniiii. + lb f + 3 Tablets +2) ° T.ue Alt. T'39 True Alt. 5 37 Su,.p. Z.n. D.st. 93 39 Sup,.. Z.n. Dist. 95 37 Deil. 22 ji N. D.lI. iS o.S. LA.11...K 71 8 N. LAr.T,i.K 77 37 S. blunder was di.sc.ivf red to have been made, but the cxii-tenre of an error in the latitude wni euspected only Irom the circun.Btanec of the ship Leing beset w.th ice. In crosiing the meridian of lob", when the long, changes from W. lo K., or from K. to \V., eare mu»i be taken to change ili<! application of the i<irr. of the declin accurdiiigly. The neglect of this prc-caut:uo has been a fertile &ource uf uiiiitaUc^. 246 NAUTICAL ASTUONOilY. C85. Accurately. Reduce the declin. to the nearest second for the long., correct the refraction for the harom. and therm, and add the sun's parallax. As the sun passes the meridian at 0'' C" 0' App. Time, llio Greenwich Date may be deduced in App. Time by means of the Ion", in time, No. 576 (3). Or it may be taken at once from tlio chronometer, in which case it will be in Mean Time, as is supposed in Ex. 1, following. 5' K.. Ex. 1. March Jotli, 1878, long, 1° 25 W., obs. mer. alt. Q in the mercury 69° 8' 10' bearing S., rime by chron. lo'i oil ijiii ,2., inde.\ error + i' 10°, bar. 29-5 inches, therm. 40°. Q's Decl. 20th 0° 5' 38'-7 S. 2ISt o ig 24 N. Daily Var. I3'"i2-,var. 23'4i Re.l. Decl. Oba Alt. Ref >'2S' •riu r. + Bar - > Semid. [•ar. True Alt. Z^n, Dist. Decl Lat. 23 4" I 5 3« 26 6 ■7S. 5 ■I « 69° 8- + I 10 2)69 9 10 Ex. 2. June 20, 1S78, long. 26° at midnight, obs. mer. alt. Q in the silver 26° 26' 20", index o', bar. 29*8 therm. 34°. Green. Date, AT. .Tune 20'' 10'' 15 Reduced Decl. 23=27' Obs. Alt. True Alt. Supp. Zen. Dist. 103 25 7 Decl. 23 27 16 N. Lat. 79 57 51 N. Ex. 3. July 27th. 1878, long, obs. mer. alt.0 in tliequicksilver 1 16" zenith N. ind. corr. +2' 15". bar. 30-1 therm. 60" ; required the Latitude. Green. Date (A.T.). 27'' o" 8'" Decl. 19° 12' 17' N. ; True Alt. 57' Lat. 51° 26' 13' N. 26° 26' 20" '3 '3 .0 -3 57 '3 n 46 + 13 25 7 ; Red. 46' 4' : 686. When the altitude of either limb of the sun is observed, and the alt. of the other limb (which will appear tlie same in tlie instrument) is observed fi-om the oj)posite point of the horizon (No. 5.35), take half the diif. of these angles and add to it the correction of alt. ; tlie siun is the true zen. dist. Ex. 1. Aug. 5th, Obs. Alt. Q N. iS7S,long.2 5°W. ii5"46'-3 63 49-3 Ex. 2. Oct. 2Cth, Obs. alt. Q N. Diff. Corr. of Alt. Z™. Di.st. Red. Decl. Lat. 1878, long. l°W. 105° 5' 74 32-2 Diff. i' 57 30 3- •* Corr. of Alt. 25 58 -5 N. + '4 .5 .f._-+N. Zen. Dist. Red. Decl. Lat. 25 58-9 N. 16 56-3 N. 42 55-1 N. 15 .6-6 N. 10 23-9 s. 4 51-7 N. Thus it appears that this observation, which is the most ciliciont in practice, is also the shortest in computation. Kx. 3. July 15th, 1878, alt. Q N. 93° 58', S. 85° 38', long. 71° W Ez. 4. Julj4tb, i8;S,alt. N. ST 59', S. 97" 4' ', long. 83° E. Lat. 2 5^3 9'-7N. Lai. 15 y 7 N. FINDING TIIK I.AHIUDE. 2 17 9. Meridian Altitude of a Star or a Planet.* 687. 'ITie Observation is the same as for tlie sun, but it is still more necossary to take separate altitudes of a star in order to avoid straining the eye to jierceive its small rise or fall when near tiie uieriilian. See No. 542. 688. The Computation. At Sea. (1.) Take tlic det-l. fitJu-r from the Nautical Almanac, or, in the case of a star, from 'V\\\<\v <i3. (2.) Correct tiie alt. for index-error, dip, and refraction, No. 6.52. Find the zenith dist. and proceed as for tiie sun. Ex. I. IMay Kth. 1S78, ohs. imr. alt. of [ El. 2. April 5th, 187X. p m. Inn- iif Spica 33° 17' S.' imlex f rror + 1' 2o', cvu ' W., obs. alt. of Mars 4>/ 20 rs., ■■nli < ^ofcet. ' ; corr. + 3', eye i6feL-(. Obs. Alt. 33^ 17' S. ' " N A. VY ^44; tl e M.T. ol' .ncr. ,.as.. ( ,. j DaK- is Aug. 9"' la'' 0°, ami the Red. Di-cl. Ref. -I ! Trm-Alt. 33 11 S. Ziii. nist. 56 48 N. Suir's Uecl. 10 52 S. L.\T. 46 16 N. is 23° 39' N. Obs. Alt. 49°2o'N. In.lesCorr. +3') Dip -4 '. Ref. - I Red. Decl. 23 39 N- L.\T. 17 3 S. Ex. i. Dec. 2ist, 1878. obs. mer. alt. Aldebaran 50° 27' N. ; heiglit of eye 20 feet 1 required the Latitude. L\T. 23" 22' S. lix. 4. Jan. 1st. 187S. obs. mer alt. Sirius 81^ 13' S., iud. corr. —4', height of eye j8 feet: rccjuired the Latitude. Lat. 7° 38' S. F.x. 5. Feb. iSth. 187S. obs. mer. alt. Canopus 37"" 25' .S., ind. corr. +2', height of eye 16 feet: required the Latitude. L.at. o" o' Ex. 6. Feb. 1st, 187S. obs. mer. alt. Arcturus 80° 12' N., ind. corr. + 4', height nf eye iS feet: required the Latitude. Lat. 10° i' b. Ex. 7. Feb. i8th, 1S78, obs. mer. alt. x Lyrie. below tlie pole, 12° 30', ind. eorr. + 2', height of eye 18 feet : required the Latitude. Lat. 63° 44' N. Ex. 8. Oct. 6th, 1878, long. 87° W., obs. mer. alt. Mars 57° 45' S., index corr. -2', height of eye 18 feet. Ex. 9. July 6th, !878,long. 178° E., obs. m height of eye 20 feet. Ex. 10. J.in. 6th, :878, long. 169° W., obs. height of eye 15 feet. 689. Accurately. Take the decl. from the Nautical Almanac. For a planet find the (Jr. Date, and reduce its hor. par. aiul decl. Correct the refraction for the thermometer and barometer. 690. Stars which never set at the phu^e may be observed botii above and below the pole. In this case the latitude is half the sum of the altitudes corrected for refraction. 691. If two stars are observed on the meridian, on dilferent side3 of the zenith, and at equal altitudes, the result is independent of the rc'fraction, unless it changes in the inteival of tiie obserxations. If the altitudes arc not equal, the le.suit involves only tiie ditlerence of the refractions j)roper to each. Lat. 30° 15' N. dt. Jui.iter 57°5o'S., indexcor Lat. . 5 ■ +5'. r. alt. Venus 69° 54' S., ind.- X cor Lat. 9" 15' n' * Veiiua may oili^i be observed bf di\yiight, even in high 248 NAUTICAL ASTRONOM* 3. Meridian Altitude of the Moon. C92. The Observation. Tlie same as for tlie sun. See No. 540. 693. The Computatum. At Sea. (1.) Find the Green. Date by means of the time at ship; or, if tliis time is uncertain several minutes, find the M.T. of tlie moon'st mer. pass., No. 627, &c. Reduce thereto f,he moon's decl., No. 589, her hor. par., and take the corresponding Bemid. from Table 40, all to the nearest minute. (2.) Correct the observed alt.. No. 654, and proceed as for the sun. No. 683 (3). Ej. 1. Nov. 3d, 1878, lonR 150° 15' E., It 7'' 7" P.M. mean time at ship, cibs. alt. y^ 45° 13' S., height of eye 16 teet. M.T.S. Nov. r' 7' 7° liOng. in time — IQ ' E. M.T.G. Nov. "i 21 6 >'s Decl. at ii' 6", var. 1 1 9" Heil. Decl. Hor. Par. Semid. Obs. Ah. 1 J)ip -4"l, Si mid. +15/ ■4" 47' 45" S. — I II '4 46 34 S. 43 58 N. 14. 47 S. 2J 11 N. Ex. 2. May icth, 1878, a.m. 1<.m^'. W., ol)s. mer. alt. "j" 48° 48' S., IilI. e>e 18 feet. Moon's Jler. Pass. 19'' 15' M.T. Mer. Pass, at Long, in time M.T.G. May rs Decl. at as" 4", var. 69" Red. Decl 19 15 7_ 19 23 56' 33' 15 26 Ohs. alt. Dip Semid. .i6j 30' and H.l True Alt. Zen. Dist. Lat. «.^'. 1 34 a'». loth, 1S78. r.M , alt. j 59" 44' N., iiKlex corr. of eye 18 fett. Lat. 53° 4»' S. t:%. a. Dec. 2ist, 1S78, A.M. lon^'. 149^ I Lx. 4. A W'., obs. mer. alt. J) 84° 9' N. index eorr. E., ohs. me + a', height of eye 14 feet. - i', height Lat. 31° 14' S. I It will in general be loss of time to work nearer than to mi-nitei, because the moon's declination cannot be found to seconds unless the GrL-enwich time is known with precision.* 694. When both the upper and lower limbs are well defined, the BU]ip]. of the alt. can be observed, and the precept No. 683 applied. When only one limb can be observed, the semi-diameter must be ap])lied. 695. Degree of Dependance. The error of the resulting lat. is obviously the sum or dittei-iiicc of tlio errors of alt. and decl. The hit. by the sun at sea may be i1c|h"iu1i.m1 ii|hiii within 2' or less, thai by the moon not so nearly, ami the lat by a single star in a dark night perhaps not within 3' or 4'. • .Also as the moon at certain time.s changes her < h»r mer. alt. may differ considerably from the max aiiuutes may occur between these two altitudes Sc< FINDING TUK LATITUDE. 219 Errors of ob-iorvation or of tlie instrument nuiy be reniov(<I h fniploying celestial bodies of nearlv equal altitudesN. and S. of the zenith.* (See No. 999.) It may in general be considered that the lat. by iiier. ait. is not decisively dc-tcrinined unless alts, on both sides of the zenith have been employed. II. By the REI)L'CTl{)>f TO THE MeKIOIAN. 696. When the sky is cloudy, or tlie weather variable, the sun or any other celestial body, though obscured wiien exactly on the meri- dian, frequently appears, for short intervals of time, both before and after the meridian passage. t \\ hen the body is near the meridian, tlie change of alt. in a small |)ortioii of time is very small ; and thougli the altitude near tlie uK^ridian changes at a different rate in different latitudes, yet the change of altitude in a given small interval is not sensibly affected by a change of several miles in the latitude, and therefore it may bo computed with tolerable accuracy, even when the lat. by account (which is used in the computation) is considerably in error. If, ac- cordingly, at the time of observing an alt. near the meridian, we know the hour-angle, we may find very nearly, by computation, the difference of alt. by which to reduce the observed alt. to the nier. alt., and which is thence called the Reduction to the Meridian. This method is, in ))oint of simplicity, but little inferior to the meridian altitude, to which it is next in importance ; and it particu- larly demands the attention of seamen, because, when the latitude by observation is left, as it too generally is, to the casualty of obtain- ing the merid. alt., it is frequently lost for the day. 697. The term "near the meridian" implies a meridian distance limited according to the lat., tiie decl., and also the degree of precision with which the time is known. The Limits are given in Table 47 See also Explan. of the Table. 698. Since the lat. by ace. is employed in com])nling tlie Rciinc- tion, it may be necessary, when this lat. has been tbiuid to be much in error, to repeat the work. * Though the lat. by a single star may not be very correct, yet the error will in genrnil Be much less than that of the D.R. The altitude of a star also atlorils a certain click against the mistake of ajiplying the sun's declination the wrong way ; and it may be remarked, that a single observation of the kind would have prevented all the delay, wear and Icar, and danger incurred in the cases mentioned in the note p. 244, from the ships being so far out of their proper latitudes. t Capt. Sir Richard Grant remarks that in H.M.S, Cornwallis, alts, of the sun ind Hars were rarely to be obuined while within the limits of the Gulf Stream, but they bad a nmmentary glimpse of the sun near noon once in two or three da^B. — Ncutical Magaiiue, 1H3H, p. 4a:. ^OO NAUTICAT- ASTllONOMT. 1. Reduction to the Meridian at Sea. [1.] Bulhe Sun. b99. T/te Obskvvation. When the sun is witliin the liniits in Table 47, observe two or three altitucies,* quickly, noting- the times. When the alts, are not observed very close together, eitliei- a separate result slionld be obtained from each alt. with its corre- sponding time, or the case should be solved by No. 717. 700. The Computation. (1.) Take the mean of the alts, and the mean of the times. (2.) Find the sun's hour-angle, or the time from noon, thus: 1 . When the App. Time has been lately determined ly observa- tion. If the ship has since made westing, subtract the diff. long, made {rood from the A.T. found ; if she has made easting, add the diff. long, to the A.T. : the result is the A.T. required. 2. When the A. T. has not been lately determined by observa- tion. Find A.T. by the chron. and the long, by ace, thus: To the G. M.T. (found by applying to the chron. the gain or loss up to the time) reduce the JEq. of T. and apply it to the G.M.T., as directed page II. of the Nautical Almanac, or the contrary way to that directed in Table 62 : the result is A.T at Greenwich. In W.low^. subtract the long m time from this Gr. T. (increased, if necessary, by 24''); in E. long., add it : the result (rej'scting 24'' if it exceed 24'') is A. T. at ship. When the A.T. of observation is p.m., it is the hour-angle re- quired; when it is a.m., subtract it from 24'': the rem. is the hour- angle. If A.T. is near 12'', subtract it from 12''; if it exceed 12^ reject 12'": the rem. is the hour-angle from midnight. Find tlie sun's decl.. No. .079. (3.) Correct the alt., No. (J47. (4.) Add togetlier tiio logarithm from Table 70 and the lo--. sme square of the hour-angle: the .sum is the log. sine of the Reduction. {ft.) Add the leiliiction to the true alt., unless the observation is near midnigiit, mIicm suhtruct it: the result is the mer. alt. at tho place where the alt. was observed ; and the resulting lat. is the lat, of the ship at the time of observation (not at nooii). Having the mer. alt., proceed by No. 683 (3). Ex.1. Aug. 5tli, 1S26. H.M.S. Leven. lat. by ace. 47" N. ; long, by ace. 25" \V. at 1 1'" 48" before noon; obtained true alt. 63° 54' to tlie soutbwaid ; reijuireil the Int. The reduced decl. was 17^4' N. Mer. alt. 64" o' Lat. 47-, decl. 17 ' [same name) II'" 48" sine sq. Red. 0° 6' sin. 63 54 Mer. alt. 64 o Zen. dist. 26 o N. Red. decl. 17 4 N, Lat. 43 4 N. Heiieatiug the work gives 43'^ 3' i more than one altitude would, for greater security, always be 1 shall, to avoid repetition, consider the term "altitude" in the : i, ua iui{>Jyiivg the mcau of two or more altitudes corresponding Fx 2. Ut. ^,'6'N.,Os.lc-cl liiiiiJ tlie Lutitude. The Red. FINDIXO THK I^TirUDE. i dccl. jo° 4' S., at oi' 54"' 12' I'.i 54', raer. ah 14° 55', : 2.51 sun's true alt. 14 tlie Latitude 55 Ex 3. Feb. 13(1, 1878, lat. by ace. 40° 5' S., long. 132° E., at II' 45" 20" t.K., oIm. alt Q 59° 40' N., iiidix corr. —2', eye lo IVet : find the Latitnde. lied. decl. 9° 54' S., true alt. 59° 49', Red. 1 1', Lat. 39° 54' S. K.I. 4. Dec. I2th, 1S7S, hit. by ace. 0° o', lon.i^. i6i° W., at o" II" 52' t.m., olis. alt. Q 66° 34' S., index eorr. —5', eve 16 feet: required the Latitude. Red. decl. 23° 7' S , true alt. 66° 41', Red. 11', Lat. 0° i' N. Ki. 5. June 21st, 187S, hit. by ace. 42° 18' S., long 53° E., obs. alt. ^23° 4'' N. Inde\ eorr. - l', eye 14 I'lel ; lime hy natch o* 50" 53* p.m., f.ist on A.T. 14" l8', ilill'. long made suiee 20' K. : hnd the Latitude. Red. duel. 23° 27' N., true alt. 23° 50', Red. 35', Lat. 42° 8' S. 701. When the number of minutes of arc, in the Refhiction, exceeds tlie number of minutes of time from the meridian, it is proper to refer to Table 48, to ascertain if it be necessary to eini)loy tlie Second Reduction. Ex. 1. (The preceding.) The number of min. in the Reduction, or 6, being less than the number of min. of time, or 1 1, it is not necessary to refer to the Table. To Compute the 2d Red. Double the loij. sine of the Red. ; add to it the log. tan. of tiie nier. alt. found, and tlie constant 9*6990: the sum (rejecting tens) is the log. sine of the 2d Red. This is to be subtracted from the 1st Red. (above the Pole), that ia applied to the alt. the contrary way to that of the 1st Red. Ex. 2. May 5th, 187,', lat. ace. 5" 3 N., long. 71° 10' E ; time hy natch 5'' 3" 7' p.m., t on app. time at ship 4'' 47'" 27': obs. alt. Q 77' 59' N ; height of eye 16 feet. Time by Watch 5'' S"" 3"' 7* Lat. 5°, nee 1 i6i°(»ame> a,,,,-) 0-992 Fa<t -4-47 27 0'. ,5 " 40' sin sq. 7-067 A T.fi. 5 15 40 39' sin. 8os9 Lun-. ill Tim -4 44 40 True Alt. 7S 11 A.T.(;. 4 19 31 6 1.8 n.ci. 1(1- r\. C.rr. ^ .+ Ohs Alt. 77' 59' Tahle:iH H-,1 78 50 tan^. 0-705 cunst. 9-699 Sin. 6-522 R«d.Uecl. lb ,5 N. True Alt. 78 11 . Mer AU. -S 49 Zen Dist. 11 1 ■ s. Decl Lat. ,6 ,5 N. 5 4N. Ri. X .Inn. fith .8-S, , 3" 36'" 28" sl.nv on AT., h lo- N., long. 58° E., at S" 4"' 5;"hy niieh. 13' W. ; obs. ait. Q 65° 13' S, height oC cya , Red. 31', 2d Red. o', Lat. i ' 34' N g. 110° W. at o'' II'" 19" r.M. AT. S Sept. i^th, lS-8. lat. ace. 4° 58' S.. lor Si-" 33' N., index error —2', eye 16 feet : find the Latitude. Red. decl. i" 52' N., Red. 30', zd Red. i', Lat. 252 NAUTICAL ASTRONOMY. 702. If a second altitude, some time after tlie first, do not confirra the lat., the time is probably in error. In such cases the mean lali- lude is not to be taken as the true latitude, because tiiat result which is nearest to the meridian is the best. If the time only is in error, it will be easy to find, by trial, thai time from noon which will make the two results agree ; and tims this observation may serve to correct, approximately, the error of the watch. When the interval, however, between the alts, amounts to 6"' or 8"", the case should be solved as a Short Double Altitude, No. 720. [2.] By a Star, a Planet, or the Moon. 70-3. Compute the hour-angle: this must be done by means of the time at ship, by No. 611 or 612. But in general it will be better to observe the alt. of a star nearly E. or W., and to deduce its hour- angle, as directed in No. 737. In other respects proceed as above directed. When the decl. exceeds 24°, the log.. Table 70, must be computed. 704. Degree of Dependance. The error of the result is composed of that of the mer. alt.. No. 695, together with that of the com- puted Red., which latter, when well within the limits of Table 47, will rarely be worth notice. 2. Circummeridional Altitudes. 70.5. On shore, when the time is accurately known, or even at sea under favourable circumstances, the result of several altitudes may be obtained by a computation which is the same in principle as the preceding, and is of much greater value than that of any single observation on or near the meridian. [1.] By the Sun. 706. The Obse>-vation. When the sun is within the limits in Table 47, observe altitudes as fast as convenient, noting accurately the times by watch, of which the error on Apparent Time must be known or found as soon as possible afterwards. When precision is required, note the barometer and thermometer. 707. The Comi>utation. (1.) Find the Green. Date for noon at the place, in apji. time, and reduce the decl. If the error of the \^atcll is given on M.T., reduce also the Eq. of Time. ';2.) By means of tiie error of the watch obtain A.T. at each altitude. To these App. Times take out the Reduction in seconds IVom Table 49. Take the mean of the Reductions. v3.) Find the mean of the alts., and correct it. No. 649 or 650. If the meiidian alt. is not observed nearly, deduce it. No. 663, &c. (4.) Add together the log. of the mean Reduction, the log. cos. rjf the lat. bv ace, the log. cos. of the decl., and log. sec. of the uier. alt.: the sum In the log. of the Rohirlion. (.*>.) .Al niioii, ii/ld llic Reduction suhtrurl it: llic rcsull is tiie iii'.T. iilt HF. LATITl'liR. \h- hi 2,-,3 Julv 9<li. iS-(6 lilt by ace. 51° ...-ur lux.n by a se ■^. by Wiiirh. D„uWe Alt. Q .jg,,,,,. ,20" 28' ^ 5 o 47 120 30 30 5 3 40 120 32 37 . 25 46 120 7 49' N. ; long, o'' 3"' W. ; obs. alts, of the At ii"" 55"" I' by waU'h tile wateh was 2"' io"'9 fast on M.T., and at c^ ^^^" (f it was 2'- 8-7 fast. n.e observation being at uoou in long, o'' 3'" W., the Gr. Date is July gtli, o" 3"', ap^ rbc reilueed Et]. of T. is 4"' 49" 4, sv/z/r. from M.T. ; red. deil. 22° 21' 1 1 " N. Errnroii App.T. T.byW. ii'^ss- .■ App. Times R«lucli<ins "? Lat. Deri. M. Alt. 472"-4 log. 472" Mer. Alt. Zen. Dist. Declin. 60 29 COS. see. 7' 24 27 M.T. 11 5^ 50 Kt,. of T. - 4 49 18 46 69, •. App. T. II 4^ I T. by watch 1155 i 5)2032 -9 406-6 60° Ref.-.. 33" Par. - 4 Mean Corr. 29 Th. 61°, ) Alt.6o°,-o"-9 Bar. 29-8, -0-^) -i True Corr. 28 60 2 s 10 W. fast on A.T. 7 Sum of Alts. 601° 29' 47" 120 17 57 2-609, 9-7911 9-966, C-3079 2-674J 4^ 60 9 25 Mean Alt. 60 25 10 True Alt. 60 24 42 II N. Approx.Mer.Alt. 60 32 Lat. 51 48 37 N. 708. To compute vhe 2d Reduction. Take from Table 60 the "id Reductions (these will be sensible in the larger hour-angles only), and divide the sum by tiie wiiole number of altitudes. To twice the sum of the three logs, used before (namely, hit., decl., and alt.) add the log. of the mean of tiie 2(1 Rodiiclioiis ; the sum is the log. of the 2d Red. required. Ex. (Ex. 1 ) 23"'39' i8 46 2d R. 3 logs. 0-0651 5) o"'30» o -S log. 9-9031 2d Red. i"-o8 log. 0-0333 Subtracting i"- 1 from 7' 52" -4 gives the lat. omitting decimals, 51° 48' 38". 709. ^^'lleu tlie declin. changes considerably, take the difference between the sums of the Eastern and Western lioiir-augles, in deci- imil.s of an hour; niultijily it by the hourly diff. of declin., and diviile by the number of altitudes. When the sun is upproarinng the elevated pole, if the E. sum is the fireiiter, mid this quotient to the Red. ; if the lesser, siihtract it. When llie sun is rccvilimj from the elevaled pule, the rontrnry. 254 ISAUTICAL ASTRONOMY. Ex. 2. May 7tli. 1S47, lat. by ace. 55° I'N.. long, o'' e™ W., obs. alt. of sun's aliemat* lbs in tlie quicksilver, near noon, witli the circle; bar. zg'*) inch, therm. 52°. During the observation the angle was cai'ned twice quite round the limb, and the final angle registered was Increased by TotaJ Angle ( The error of watch at 1 termined by equal alts , f<u:t on A. T. which gives The obs. being made a long, o*" 6'", the Green, Da 7th, o' 6™ in Aiip. Time. Bum of Alts. 16) 1651° 59' 30 App. Times I," 36-2.. Sun's Decl. at Green. Date, i6"43' i "N. To find Appro-x. Mer. AJt, Decl. 16° 43' 90 106 43 Lat. - 55 I Mer. Alt. 51 42 ductioni Sd Hed. 16)9 To find the Effect of a Change of Declin. The Sum of the E. H.-ang. is 97"' 30" Do. Western do. DiH'. of E. and W. H.-ang. •13 94 53 •' 37 ) zi 30 ... 1083-3 •• Sum 58i7'4 D0.-H16 363'6 log. 2-5606 o -6 log. 9-7781 Lat. 55'" I' COS. 9-7534) Decl. 16 43 COS. 9-9812 [ 9-9474 Mer. Alt. 51 42 sec. 0-2078 ) 2 322-2 log. 2-5080 9-8948 9-8948 322"= 5' 22' 2d Red. 0-5 log. 9-6729 Alt . 51 36 4 9 Mer. Alt. 51 42 11, and Lat. 55° o' 50" Eliect of Change, Decl. is o"-i3 only. 710. The rate of the watch must be allowed for in deducing each hour-angle. In tl-e case of the sun the rate should be found upon A. T., but it is ol course near enough for this purpose to employ M. T. 711. An error in the absolute time affects all the liour-angles alike, but it produces the greatest errors in the greater Reductions. The higher the altitude, the greater is the precision required in the time. When the time is inaccurate tlie Reductions on one side of the meridian will be too great, and on the other too small ; if, therefore, the altitudes p.m. be taken so as to correspond nearly with those a.m. the errors of the Reductions will very nearly compensate. This distribution of the altitudes, by equalising the number of the hour-angles a.m. and p.m. has also the advantage of neutralising the effect of a change of declination. It is proper, moreover, to multiply the observations near the meridian, in order to weaken, by subdivision, the small errors to which tlie outer reductions may be liable 712. The effect oi irradiation, or the increase of the sun's appa- rent diametfr caused by the extreme brightness, and which may amount to 5" or 6" (Dr. Robinson on Irradiation, " Mem Roy. Ant. Soc." yol. iv.), is removed by observing both limbs. rlNDINT. TIIIC LATITUDE. 255 [2.] By a Slar or n Planet. 713 The Ohservation is tlio same as for the sun, No. 706. 7!4. TItc Comjiiitfitinii. (1.) Haviiifi" tlio error of the watcli on M.T., hnd tlie Greciiwicli Date, llediiec tliereto the Sidereal Tiii>''. »i mean noon, and also the II. A. and decl. ; and for a planet, the lior. j>ar. ('2.) Find tlic lionr-ansjle at each alt. and inocecd as for the sun. When the watch shews Sid. Time, the hour-angles are obtained at onee. 715. The stars near the poles, and espe<'ially the pole-star, are the i)est adapted to this observation ; hccanse, fimn the slowness ol UK! motion m altitude, an error oi' time piotiuces but little error in the Reduction. 716. Errors of altitude, of whatever kind, are removed by em- ployinij two bodies on opposite sides of the zenith, ant' at equal ■iltitu<les. A single result, even tliouirh obtained with the circle, and without the root', cannot accordingly be considered definitive when extreme precision is required. 717. Therefore, in the northern hemisphere the best south stars to pair with Polaris are those whose meridian altitudes are about the same as the latitude of the place. Similarly, in taking Lunars, stars lying at about equal dis- tances, east and west of the moon, should be chosen. See No. 8G1. III. By Double Altitude of the Samb Bodt. 718. Two altitudes, of the same or different celestial bodies, with the interval of time between them, constitute an observation which is called a Double Altitude.* The interval may extend from a few minutes to several hours. See Sumner's Method, No. 1009. 719. \\ hen a double altitude of the same body is taken, the precepts below will be convenient in directing the method of solution jiroper for the case. Also, when a first altitude lias been obtained, tlie observer will find, on referring to the numbers indicated, under the heads Ohstrrn- tion and Limits, instructions how to complete the observation in thr manner adapted to the circumstances. Selection of the Mettiod of Solution. WTion 4oM /ilfs. are not far from the meridian, on the same side, No. 729 ; on different •ides. No. "31 ; in a doubtful case, No. 728. When one all. is near tlie meridian, No. 7.S7. When neither alt. is near the meridian. If the Int. hy ace. is not Rreatly in error, No. 71fi. If it is greatly in error, or if it is proposed to do without it, No. 7.')7. * This is the old-established term ; it is, however, defective, inasmuch as the wonl doublt means twice the same. Since the process involves two altitudes used in conibiniition with (Mie another, the term which would naturally suggest itself is Combined Attitudes: we should tlien have, accordingly, combined altitudes of tlie same or different bodies, am! of long or iliort intenals. This term, therefore, which is accurate as respects denni'ion, would be clea- and dcbcriplive in use. All changes ir. nomenclature, in this subject, however, must be ui«d8 256 NAUTICAL ASTRONOMY. 1 . Short Double Altitude. 720. Wlien tlie time is not known with some degree of pronsioii, t).e Keduction to the meridian cannot bu computed. In such cases recourse must be had to two altitudes separated by a short interval, and not verv distant from the meridian. 721. 'I'lic chaiij^e of altitude in a small interval of time (No. G(JG) depends ( liittiy on the hour-angle or meridian distance, aTid is nearly I he same for a considerable ditFerence of latitude. Although alti- tudes at sea are always more or less uncertain, yet difference of alt. niiiy often be obtained with much precision. If, therefore, the dif- ference of alt. in a small interval of time be measured by an instru- ment, the hour-angle corresponding may be found by computation. The Reduction to the meridian being then computed for this hour- ungle, the latitude is obtained by the method in the last section. 722. The error of the watch is immaterial, but its rate should be known nearly enough for measuring the interval without much »!rror. 723. When the altitudes are observed at different places, it is necessary to allow for the ship's run in the interval. 724. Since the lat. by ace. is necessary in computing the Redue- iion, the work should be repeated when this lat. is found to be very erroneous. 725. Limits. When both alts, are taken on the same side of the merid., if the outer alt. fall near the limits in Table 47, the Interval should exceed one-fourth of the time of that alt. from noon, and should not be less than 5'". The observation may be comprised within double the mer. dist. implied in Table 48. When the alts, are taken on different sides, the Interval may vary from 5"" to twice the liuiit in Table 47. [1.] By the Sun. 726. The Observation. Observe an alt.* and note the tune. Note the sun's bearing for the purpose of allowing for run. After the proper interval, No. 725, observe the second alt. and bearing, noting the time. 727. The Compvtation.f (1.) Subtract the first of the two times from the second (increased if necessary by 12''); the rem. is the In- * Two only, or at most three, altitudes taken in quick succession would be employed in ohaervations with a short intei-val. t The first work in which a method occurs of finding the latitude by two altitudes observed near the meridian (but restricted to the same i-ide) with an interval of a l'c» minutes, is the " Cours d'Obsertations Nautiques," by Ducom. Tlie advantage which Admiral W. Owen acquainted me that he had derived from the practice of this method led me to give an account of it in the '* United Service Journal," vol. x., together with a rule for adapting it to longer intervals. Soon after the account appeared, Commander Graves, commanding H. M. surveying-vessel Mastiff, was enabled, as lie informed me. hy this observation, to run direct for Malta twfore the coming on of a grecate, or N.-K. gale, to wliich another ol Her Majesty's ships was exposed. FlNDINfl TIIF. LATITUDE. 2r)7 rcrval. Iteduce the declin. for the time of tlie alt iii'inosl tlie iiior,. No. 573; or to tlie middle of the interval (that is, to noon) when the altii. are equal. (•J.) Correct the altitudes, No (548 or (149. Also coirect tlie Iiitt-rval by watch for the rate, if this is very large. When the sun is rising or falling at both observations, proceed by Case I., No. 729; when rising at one observation, and falling at the other, proceed by Case II., No. 7.'il. 728. ^Vllen sufficient time is not afforded to perceive the rising or falling of the sun, and wheii it is not known otherwise whether the altitudes are taken on the same or on different sides of the meridian, proceed thus : Consider the interval* as a time from noon; and compute the Reduction to it; then. If the Reduction is less than the ditf. of alls., the observations are on the same side; if the Reduction is the {jrcuter, they are on different sides. Hence, if the Reduction is equal to the ditf. of alts., one of the alts, is the meridian altitude. No great precision is to be expected, as the rules are only approximate. In a doubtful case use either. 729. Case I. The observations on the same side of the meridian. (1.) When the alts, are both a.m. reduce the 1st to the place of the 2(1, No. 661 ; when they are both p.m. reduce the 2d to the place of the 1st, No. 662.t Find the diff. of the alts, and their mean. Correct the diff. alts, and the interval by the Table, p. 223. (2.) Add together the log. sine of the diff. of alts., the log. cosec. o."" ihe interval, the log. sec. of the lat., the log. sec. of the decl., and the log. cos. of the mean alt. : the sum (rejecting tens) is the log. sine of the honr-angle, approximately, at the middle time between the two observations. (3.) From this time subtract half the interval: the remainder is the time from noon of the altitude nearest the meridian. (4.) To this time compute the Reduction, which a])ply to the alt. nearest the meridian, and proceed by No. 700 (5): the result is the latitude at the time and place where the alt. nearest the meridian was observed. J * If is proper to remark here, that the interval l)etween two observations of the sun nhouUI, in strictness, be measured in apparent time, instead of mean time, which is sliewii by tlie watch. To correct the internal on this account, find the cliange of the Eq. of T. for tlie inter\-al. When the Eq. is additive, if it is increasing . rubtract the change ; if decreasiiiij, orfdit; and the contrary when the Eq \s mbtrttctive. In the short double alt,, however, this correction is insensible, and in long intervals the resnlt is of so inferior a kind that the trifling accuracy gained by this process can rarely be worth the trouble bestowed upon it. t This reduction is of particular consequence in this observation, because the accuracy of the result depends on that of the difference of altitudes. ; 'I'his observation, which affords the latitude, the app. time near enough for common purposes, and thence an apprcximate long, by chroiiometcr. with the azimuth (No. C78), and i»iu<e<)uently the variation of the cmnpass. will, it is presumed, be found one ot the most uxful observatioui that laii be made at sea, espeLiallj ill high litltiides. S 258 N.VUTIOAI. ASTRONOMY. El. 1 Oct. 9th, 187S, A.M., lat. ace. 34' 55' N-. long- *' height of eye 16 feet, ind. corr. + 3'. T.byWalen ii» i2'"52' I Alt. Q 46= 47' 50" Ditto II 43 4|In'i-™"'- + 3 Alt. Q 47 57 Interval ^^TTT | Table 38 _+_m_ .3 Half Int. 15 Decl. noon Corr. 61° W. Red. Decl. 5° Greater Alt. VV., had following (As.- 48° n' o Mean Alt. Diff. Alt. 47 I 50 95 .2 50 47 D. Alts. Decl. Alt. meai Mid. T. i Int. T. fr. noon 30"' I2» 34° 55' 47 36 29™ 8' '(of the greater alt.) Lat. 35°, decl. 6i°, Ta' Greater Alt Mer. Alt. Red. Decl. Lat. 8-3036 8S14 0862 48 II 48 -9 4. 4'N. 35 isN. (The Red. for the interval 30"' 12' is 37', which being less than 69', shows the observa. tions to be on the same side of the meridian, if this were doubtful. No. 728.) The zd Red. is not worth notice. Repeating the work gives 35° 18' N. Ex. 2. Aug. 4th, 187S, lat. ace. 41° 54' N., long. 39" W., obtained true alt. © 63° 5 7- 5; after 11"' 12" true alt. 64° 3 2'- 5 (allowing for run). Red. decl. 17° 12' N. ; mean alt. . 42° N., decl. 17° N. o-^2~ 23'" 40' sin. sq. 7 42; 0° 31' sin. 7-952 64 33 .5'; diff. alts. 35'-o 35' o- sin. 80C7S ,,... ,,. cosec. 1-311 1 4'° 54' sec. 0-1 2S2 17 12 64 ,5 Mid.T. 29 lint. -5 'i6« 36 sec. 0-0199 COS. 9-6379 sm. 91049 23 40 (The Red. for ,,„, 2- is 6'-9, which Whence Lat. 42° 8' N. The 2d Red. is not worth notice. I 35'-o. See No. 728.) 26, a.m., lat. by ace. 47° N., long. 13° \V., obtained true alt. S -.sc E. by N. 7 knots ; after 12"' 14" obtained true alt. © 56' 37'^ 55°4i'-6, mean alt. 56° 11', diff. alts. 56-3, reduced decl. 15^23' N o™ 14'. Redu mer. alt. 58° 34 J'. Lat. Ex. 3. Aug. 11th 55° 4i'-9, bearing S., 1st alt. corrected for r Corrections, p. 205, c The mid. time fr 46° ^^' N. The 2d Red. by Table 48, alt. 58^ is i' for Red. 1° S., and therefore for Red. 1° 54' it exceeds i'. 730. Degree of Dependance. The smaller the hour-angle, the Jess is the effect of error in the D. alts. As tiie interval ma^', from its sinallness, be assumed to be correctly measured, the value of the n.'sult depends chiefly on the difference of alts., and may be estimated by finding the effect of an error of 1' in the diff. of alts., which is easily done. Divide tlie middle time by the diff. of alts,, both in minutes: the quotient is the number of minutes of error in the time from noon, caused by 1' error in the diff. of alts.: the case now becomes that of an error in tlie Reduction itself. No. 704.* Ex. In Ex. 3, above, 60'" divided by 56' gives i""i, which is the error in the time from noon, supposing 56' to be i' in error. Now, by iuspecting Table 47, lat. 47° and decl. 15°, {same name) give zj'" as the limit, or time from noon at which 1'" error of time causes 2 * Wlien the lat. is. found to have been very erroneous, repetition is very e.isily effected, B9 the sec. lit. is tlie only log. in 729 (2) that changes. FINDIXC TliK LATirrOF.. 259 error m the redaction ; hence i'"-i error at i'' fnun noon will oauae about ;' error in tlie Ueduction, and tlierelore in the latitude. This example is not an eligible one, since 12'" is only i-5th of 1'', instead of being not le^s than i-4th. See No. 725. 731. Case II. Observations on different sides oftlie meridian. (I.) Reduce tlio alts, to llie |dace of the alt. nearest the meridian. No. m\ or mi. Find the diti'. of alts. ; correct it and tlie half interval, when necessary, by the Table, p. 223. (i.) To the arith. comp. of the \og. in Ta'h. 70 add th- ]c<,'. sine of the ditf. of alts, and the loj>-. cosee. of half tlie interval : the siini is the log. sine of half the ditf. of the times from noon corresponding to the two altitndes. (3.) Subtract this half iliH, .'Voni tlie half interval : the remainder is the time from lux^n (or mcritl. dist.) of the alt. nearest the meridian. (4.) Compute the Reduction to this time, and ajiply it to the alt. nearest the meridian, and jjroeeed as directed. No. 700. Tlie result is the latitude at the time and place where the alt. nearest the meridian was observed. Ex 1. April 5d, 188, hit. by to the soutlinard, reduced to laist Times by Wi '54' 35 Fnterral Lat. 46°, decl. 5°, ar. 6^ a' X., inns;. 17' \V., tlie true alts, of the of obseivatio.i as below. Red. decl. 5^23 true alt. 49" 10' 30" a.m., or rising. 49 23 53 P.M., or falling. diff. alt. .3 23 Lat. 46°, de,-l. 5", log. 0-324 11-56' sin.sq. 6-X,. Red. 0" 5' si... 7-'5S Gr. alt. 49 24 Mer. alt. 49 ^9 „ . which pTBR the I.at. 45 54 N. 34 58 log. 9676 Ditf. alts. 13' 23" sin. 7-590 Half. int. 17'"29' cosec. i-ii8 Half diff. —5 33 sin. 8-384 T. fr. noon li 56 (of greater alt. ) Ex. 2. H.M.S. Leven, Aug. loth, 1826, lat.by »cc. 46 N., long. 15° W., obtained tnie alt. 59^57'-2; after 28'" 42* true alt. 59° 2o'-5, the ship having little or noway. Reduced deil. at 1st alt. 15° 40 N. 46° and 1 5°, ar. CO. log. 9-573 | over the \ interval (which should be the Ditf. alts. 36- 42" Half int. I4'"2i' Half ditf. 14 39 This small . the. greater) is due to the error of the method itself, which becomes ajiparent in a long interval, and it shews that the alt. 59" 5;'- 2 is very nearly the nier. alt. This gives tha _ uted \ diff. Lat. 45" 43' N. Ex. 3. Dec. 23d, 1825, lat. by ace. 8° S., observed true alts. © 74° 26' a.m. and 74° 16' n., with the interval 36'" 37". Reduced decl. 23° 27' S. 0-S32 17"' 4' 7''4^ Red. 0° 32' sin. 7-974 - 1 (Table 48.) 74 ^6 75 W The Lat. is S° 24' S. This Ex. is fa» rithont the limits, Table 47. Ex. 4. Aug. 9th, 1826, lat. by ace. 45°N., long. 15" W., a.m., obtained true alt. to" 2(/-5. After 52'" 27- obtained true alt. 60° 30'. The 1st alt. reduced for i' nortbitj made gi'od in the interval is 60° 28-5. The diff. alts, i'-^ and a half interval 26™ i6' give half diff. I9'; the Red. is 31', and mer. ail 61" i', which, with i.-duced decl. i^" 57' N., give Lat, 44' =,V N. Ar. CO. log. 9-168 10' sin. 7-464 1 3°' iS' co^ec. I -098 - ' '4 SIB. 7730 >7 4 2G0 NAUTICAL ASTRONOMY. 732. When the alts, are equal, the half interval is the time from noon. 733. Degree of Dependuncc. It would not be easy to give a concise rule for this in long intervals. The rule No. 730 applies very nearly in short and moderate intervals, using, instead of the "middle time," the time from noon of the alt. nearest the meridian. [2.] Short Dovble AUitude of a Star. 734. Increase the interval by 1* for every 6'". Take the doel. from the Nautical Almanac, or Table 63. In other respects proceed as for the sun. [3.] Short Double Altitude of a Planet. 735. Find the Greenwich Date for the middle of the interval, and reduce the decl. Find the daily variation of R.A., and deduce by Table 21 the change of R.A. for the interval. When the R.A. is increasing, subtract this change from the interval ; when decreasing, add it. Increase the interval by the acceleration upon it. lu other respects proceed as for the sun. As the R.A. and decl. of a planet sometimes change very slowly, much of the above labour is not always necessary: particular rules for all such cases would, however, be superfluous. [4.] By the Moon. 736. Find the Greenwich Date as nearly as possible at each observation, and compute the R.A. Subtract from the interval the change of R.A., and add to it the acceleration. Reduce the decl. to the middle of the interval, as also the hor. par. and semid. In other respects proceed as for the sun. As a proper allowance for a considerable change of declination would complicate the rule, the moon can be employed satisfactorily in this observation only in cases of very short intervals, and when her declination changes slowly. 2. Double Altitude, one Altitude being near the Meridian. 737. When one of two altitude.s is taken near the meridian, and the other when the body has a large azimuth, the outer hour-angle (or that corresponding to the altitude furthest from the meridian) may be computed nearly (No. 614), since it will not be much affected by an error in the latitude by account.* The difference of the hour- angles being afforded by the measured interval of time, the other, or inner hour-angle, is found ; and the Reduction being computed thereto, the nier. alt. is deduced. See Nos. 722 and 723. 738. Limits. The inner alt. must be within the limits in Table 47, and the outer angle should be as nearly E. or W. as possible. AVhen the outer beaiing is not near E. orW.the outer hour- • The UitUudo by atrount, refers of course, to the place t( FINDING TIIL I..\riTUOE. Sfil •ngle may be sensibly affected by the error of the lat, by ace. ; and if the inner hour-angle be not very small, the work may require to b« repeated. [1.] By M* Sim. 739. The Observation. Observe the sun's alt, noting the time and tlio bearing. After a sufficient interval (No. 738) observe the second altitude. See note to No. 726. 740. The Computation. (1.) Reduce the decl. at both observa- tions, either by Table 19, No. 579, or by the Green. Date, No. 580, and find the outer pol. dist (2.) Correct the interval for tiic rate of tiic watch when large. Correct the altitudes. Wlien both observations are A. Jr., reduce tlie 1st alt. to the 2d place of observation. No. 661. Wlien both observations are p.m., reduce tiie 2d alt. to the |)lacc of the 1st, No 662. VVlien one observaticn is a.m., and the otiier p.m., reduce tiie alts, to the place of the alt. nearest the meridian. (3.) Witli the outer alt., tlie lat. by ace, and the outer pol. dist., compute the hour-angle. No. 614. (4.) Take tiie diff. between tliis hour-angle and the interval: tiiis is the iinier hour-angle. (5.) With this hour-angle compute the Reduction to the meridian and ajtply it (No. 700 (4) and (5)), to tlie alt. nearest the merid. The decl. which is to be applied to tlie mer. zen. dist. is that reduced to the time of the alt. nearest tlie meridian. Ex. 1. .July 13d, 1878, Int. by arc. 5+° 57' N., long. 1° 25' W., at about 7'' o"' a.m. , obs. alt. Q 24° -^o , bearing E. by S. by compass ; 4'' 50'" 12' afterwards obs. alt. 54° 26 , course S.S.E., rate 45 knots ; ind. corr. + 2', eye 18 feet : required the Lat. at 2d obs. From S.S.E. to E. by S., or 5 pts., and dist. in interval 20'- 3 give corr. of lit. +11. Decl. 251 at nrnn 20" 4'N. .\lt. 14° 5? Long. .» -0-1 Lat. 54 57 sec. 0-24CS6 5^ 0- + 3 ( "^ ' P. Dist 69 53 cn.ec. 0027:4 1st Red. Decl. 10 7 74 51 cos. 9-4i-;2 Int. 4" 3o'" — 2 49 58 sine 9-8X404 ■ 2d Red Cecl. 20 5 Houran. 5'' o'"i4' sin sq. 9-5^546 Obs. Alt. 24' 30' Obs. Alt. 54° 26' Interval 4 3-5 12 d.cor. + 2'l ^, b. 38 +10 1 + M1 ^'5 Inn.H..an. 30 2 sin.sn. 7 (,-2 Lat. 55° Decl. 20 (iflwc 'laiiie) o'27l H 42 2d Alt. 54 39 Red. + 28 sin. 7-906 rr. forinn +■ 1 1 .54 39 Alt. 24 '53" M...r. Alt. _5^^ The Alt. nearest M er. is here the 2d.) Zen. Dist. 34 i;3 Decl. 20 i Lat. "c4 58 N N N Ex. *■ April 3d, 1878. lat. by ace 46' 7' N., long. 14° \V. at about 8" lo" ah. nh». »h. Q 26' 10', sun S.E. ; 3I' 26" 35" afterwards (corrected for rate) obs. alt. Q 49' 8' t« Ihc nonlhwnrd ; course W. ; rale 6 8 knots ; indei —3' : eye 16 feet : find Lut. at 2d obi. From W. to S K ia 12 ji's. ; 4 (it«. and aist. ij I give corr. of lat tit. - 16'. The \A 262 NAUTICAL ASTRONOMY. red decl. 5° 20' N. ; the 2d, 5° 23 N. ; the 1st alt. (corr. for run), 16° i'; 2d alt ♦9° J6'- Alt. 26° i', lat. 46° 7', and P. dist. 84° 40', give hour-angle 3'' 49"' 41'. hence ina hour-angle 23"! 6' and Red. +18', Lat. 45° 49' N. Ex. 3. Dec. 3cth, 1825, lat. by ace. 8' S., long. 6° W., at about 4" S" 16" by watch, the mean of 3 alts. Q 49° 9'-4, bearing S. 44° E. magnetic, course W.N.W. 6 knots ; at 6'' iS" 52' mean of 2 alts. Q 73° 39', the watch losing 4"5 an hour on the chron., and the chron. gaining 6*'6 a-day ; height of eye, 16 feet ; ind. corr. + 1' ; reducefl decl. 23° 1 1' S. In the interval, 2W, the chron. gained about i-ioth of 6'-6 or c'-j, and the watch lost io'*i on the chron. ; the measured interval must therefore be increased by 9'**4, and becomes z>> iox'45'. From W.N.W. to S. 44° E. is 156^ ; course 24° and dist. 13 miles give D. Lat. 1 r-9, to be subtracted from the Itt alt. Alt. 49° 10', lat. 8° r, and pol. dist. 66° 49', give outer hour-angle 2'' 38"' i6"; the diff. of this and 2'' 10"' 45", or 27" 31', is the inner hour-angle, which, with alt. 73° 52', reduction 1° 27', and 2d reduction 4', give Lat. 8° 26' S. [2.] Double Altitude of a Star, one All. near the Meridian. 741. Increase the interval by 10' for each hour. Take the ilccl. from the Nautical Almanac, or from Table 63. In other respects proceed as for the sun. [3.] Double Altitude of a Planet, one Alt. near the Meridian. 742. Find the Green. Date at each observation, and reduce to it the R.A. and decl. Apply tiie change of R.A. to tlie interval, as directed No. 73.J, and add to the interval the acceleration upon it Proceed as for the suu. [4.] Double Altitude of the Moon, one All. near the Meridian. 743. Proceed by No. 736 as far as adding the acceleration. Reduce the decl. to each Or. Date, and the hor. par. and semid. to that nearest the meridian. Proceed as for the sun. 744. The moon may be advantageously employed for this pur- pose when the Greenwich Time can be nearly ascertained, and in all cases when near her maximum declination, because her polar distance may then be very nearly computed. 745. Degree of Dependimce. The error of the inner hour-angle 13 the same as that of the outer one, which, when the body is near E. \)r W., will be very small, even when the lat. by ace. is considerably in error. 3. Double Altitude, neither Altitude being near the Meridian. 746. When neither altitude is near the meridian, the computation is dill'erent from those hitherto given, of which the object is to Hnd the meridian altitude. We shall give, 1st, an ajipro.vimnte method, the object of which is to lind the correction of the lat. hi/ ncr. ; ;iik1, 2d, the rigorous metiiod, the object of which is to find ilie latitnde itself directly, both in Ivory's form (suited to the case in \\ liicli the decl. is the same at both obs<>rvati()ns) and in a general form. 747. The orinciple of tlie apiiruxiniiilt method will easily be FINDING TlIK LATITUDF,. 2G3 nmlerstood. Siijipose the time* to be computed at each ob:=crvalioii, then, if the interval between these cominited times agrees with that iR'fually shewn by a ijood watch, the latitude by ace. (which is ati clement of the calculation of the time) is obviously correct, but if oil the other hand, the computed interval does not agree with the interval by the watch, the disagreement indicates an error in the latitude by acc.,+ the amount of which is to be computed. 748. When the correction of the lat. by ace. exceeds 10' or 15', it may, generally, be advisable to repeat the computation ; but when it is less than 4' or 5' it may be considered rather as confirming the lat. by ace. within this limit, than as correcting it by so small a quantity. See, also, Nos. 722 and 72.3, which apply to this observation. 749. Limits. An observation that is usually a substitute for a better, which the state of the weather has prevented, or seems likely to prevent, from being obtained, must be taken when it offers itself; but when there is a choice of observations, the limits are as follows: — (1.) When tiie observations are on the same side of the meridian, the difference of bearing at the two observations should e.xceed the lesser true bearing. (2.') When on different sides of the meridian, the supplement of the diff. of bearing should exceed the lesser true bearing. The dift'. of bearing should, when possible, be 90". 750. The simplest case in com])utation. This will of course be selected when the weather allows a choice of observations. In N. lat. both altitudes are to be taken to the southward of E. or W. (or the ]irime vertical) ; in S. lat. both are to be taken to the northward of E. or W. When the lat. and ded. are of contrary names, the simple case is the oidy one that offers itself", and therefors applies to the sun during the six months which include the winter. When the lat. and decl. are of the same name, the hour-angle at each observation is to be less than the hour-angle in Table 29, or the altitude is to be ijreater than the alt. in that Table. [1.] Double AlliluilenflfieSm. 7;')1, T/ie Observation. Take the alt. (see note to No. 726), noting the time, and the true bearing. After the proper change of bearing take the other altitude, noting the time. As waiting for the proper change of bearing may risk the loss of the 2d alt. it will be prudent to j)rovide an altitude earlier to serve in case of accident. * As the hour-angles only are here mni-erncd, the ennsideration of Time, as found by ■ierv'ation, will present no diffirultv to a learner. + Admiral Sir Edward Owen informed me, that when in the North .Sea he made ronstant ■ of the method of finding the lat. by the discrepaney of the computed times, as he found (iiuch more convenient in practice, in rases where it was necefsary to profit by every oppor- lity of observation, than any solution of the Dovible Altitude us a question of latitude only. Lynn's Tables the same problem is worked by trial and error. In Capt. Owen's jnurnaia : ob»trvalion, sol'cd upon the same jirinciple as that hire adopted, constantly otciim. 261 NAUTICAL ASTROXOMT. Not(! at. each observation whether the sun is to tne nortliwaid or to the soutlnvard of E. and W. An example will shew how to select the simple case. Ex. 1. Oct. 3d, lat. 25° N. The lat. is N. and declin. south, and it is the simple case. Ex. 2. Sept. ist, lat 40° N. The decl. is S° N. ; hence (Table 29) the 1st alt. must Ixi taken after 6'' 39™ a.m. (which is the suppl. to 12'' of the hour-angle 5'' 2i"'J, and the 2d before i,*' 21™ p.m. (A. T.) ; or each ait. of the centre must exceed i2°-5. 752. The Computation. The approximnte method.* If the difference of azimuth is not considerable this method should not be employed. In low lats. it will accordingly be less serviceable than in high latitudes The proper limits for ihe solu- tion will be seen on inspecting Table 71; cases outside the limits should be rejected, and those bordering on them employed with caution, especially if the error of the latitude by account is large. (I.) Find the Green. Date at the first observation. Reduce the declin. to each time of observation. For the sun, it is immaterial whether app. time or mean time be used. In general at sea app. time will be preferable, because when the observation confirms the lat. by ace. the apparent lime at ship is determined. Find the polar distances (iNo. 443). (2.) If the rate of the watch is large, correct the interval for it. Correct the alts, and reduce the 1st alt. to the 2d place of observation.t No 661. (3.) With the alt., lat. by ace, and pol. dist., compute the hour- angle at each observation. No. 614. (4.) When the oliservations are on the mine side of the meridian, take the difference of the liour-anoles ; when on opposite sides, their sum. If this diff. or sum agrees with the interval by watch witiiin 10', or even 20', provideil the difference of azimuth is considerable, the lat. is confirmed, and the time is also obtaiuevl, nearly enough in the open sea. If they do not agree, jiroceed thus: — (5.) In N lat. if the body at both observations is to the south- ward of E. or W., it is the simple case (No. 750); if the body is to the norfhwiu-d of E. or W., mark such hour-angle V. In S. lat., if the body at both observations is to the northward of * Tliis method, besides affording the time when the lat. by ace. is not very erroneous, employs the azimiiths, which in practice is a considerable advantage, since the azimuth is the means of determining the degree of dependance of tlie lat. by double altitude. f As some misunderstanding has prevailed upon the necessity of correcting the interrat of time for the change of toiigiliiilp of llu- ship, the following illustration, which was given in »n8wer to the question, in tlir Nauticid Ahi^-azine, 1840, is here inserted : — Suppose at a place A, at 10 .\.m., iIic snii'.< alt. is observed 13" 18', and 3''40'" afterwards a 2d alt. is observed. These twn alls, willi the interval 'i^ 40" afford the latitude of A. Again, suppose at a place H an olis i\n- liail nl. mined the alt. at 10 a.m., or exactly at tlie same instant the observer at A tiM.lv li]> Ui ilt., ;iiul 3'' 40"' afterwards he obtains his 2d alt. 14' 15. Tliese two alts. will, thr ini.n.,1 :,'' HI'" afford the lat. of B. Now suppo.se a ship had left A at 10 a.m., having o!.t:nii,,l il„' 1 >| alt 13° 18', and at the end of al" 40'" she arrives at B, where she obtains lier 2d alt U 15'; then she has the given interval 3'' 40"' with the 2d alt. 14 15'; and it is clear that by reducing tiie 1st alt. observed at A, or 13 18', to what it would have been if observed at B (that is, in other words, correcting the l.st alt. for the mere c/iangf ff place), she has precisely the elements for dete."mining the lat. of B, which is required. Thus, when the interval is measured by a 'vatoli, no correction for longitude appears. ttndim; the latitudk. 2C5 E. or W., it is the simple case; if tlie body is to the southwurd of F« i>r W., inarlv micIi iiour-aiiijle V. If the bearing has not been observed, or if it is doubtful, look in Table '29; then, if the computed liour-anijle exceeds the liour-aiigle in the Table, mark it V ; if the comp. hour-angle is the lesser, w'.e iio mark. If both hour-angles are less than in Table '29, it is the simple case. (fi.) For the Correction of the Lat. Compute the azimuths at each observation, No. 676. (7.) When the observations are on the smiie side, both of the meridian and prime vertical, entt^r Table 71, Part I. witii the azimuths. When the observations are on different sides, either of the meridian or prime vertical, enter Part II. To the log. from Table 71 add tiie log. sec. of the lat. by ace, and the prop. log. of the error of the interval; the sum (rejecting tens) is the prop. log. of the correction of the lat. by ace. (8.) In the simple case (No. 750), apply the correction to the lat. liy ace. according to the following directions : — Observations on tlie same side of the Meridian Observations on diferent sides of the Meridian The Computed Interval being llie greater ' the lesser The Computed Interval being the greater 1 the lesser tu6. add mid 1 «<4. In the case in whicli one or both hour-angles are marked V (No. (5) above), apply the correction according to the directions in the next Table. Observations on the same side of Observations on different sides of the Meridian the Meridian The Computed Interval being Thf Computed Interval being the greater tlie tester the greater the leaser The greater The greater Both observa- Hour /_ bei.,g Hour /. being with the with the side of the Prime Vertical. greater i texser greater lesser and both marked V. A/.im. Azim. Azim. Azim. sub. add add «/A. sub. add The Hour / V The Hour /. Y The Hour / V The Hour / V being with the being with the being with the being wi/h the difffrent sides o( the greater /mer greater] lesser greater lesser greater Ussir 1 Prime Verlical, 3t one marked V. Azim. Azim. Azim. 1 Azim. Azim. Azim. Azim. Azim. «■•«. add add s„b. add tub. .«,i. add 2G6 NAUTICAL ASTnONO^fT. Note. This second Talile, wliiih contains the remaining four'fen out of eightten rases. Diay appear complicated in its general aspect. It is, however, easy of reference when Uw case is proposed. For ex. : — 1. Suppose the observations to be on different sides of the meridian ; of this poi.it, with B long interval, there can never be a doubt. Again, 2. Lei them be on different sides of the [irime vertical, of which there cati rarely be ant doubt. 3. Let the computed interva oe the greater. Then the precept add or sub. depends on the condition that the hour -angle marked V ii *ith the greater or with the leaser azimuth. Rx. 1. (Observ. same side both of Mer. and Pr. Vert.) May loth. iS7g, lat. by ace ° ii' N., long. 6i° W., at about 8'' o'" o' a m., obs. alt. Q 35 32 , bearing E. by S. ; I ill 8'" 31" A.M , obs. alt. Q 66° 58'; index —3', eye 16 feet; course during interval S. |E.; rate 4 knots : required the Lat. at zd observation. FromS.E.iE.toE. by S. or 2I pts. and dist. i2'4, corr of 1st Alt. +11'. Decl noon, joth, 10° J' N. Alt. Q 35 31' Alt. Q 66'^ 58' Ul Red. Decl. 2c i Ind. TiibleSS +1 35 4*^ 2d True Alt. + 9 3" 9"' + 2 2d Red. Decl. 20 3 Corr. ru 1st True Alt. 35 5- 1st Honr-angle. 2d Hour.angle. Alt. 3S°5'' Alt. 67° 7' Lat. 40 12 sec V. Dist. 69 59 CO ■ 46 2 ec. 0-02706 9-46552 Lat. P. Dist. 40 12 sec. 69 57 cn.scL 77 >6 88 38 0-11701 . 0-027.5 8'3775<- 37 10 sin . 9-7.'M3 2 1 31 sill. 9-^6440 Ut 11. -angle 3° 57'" 49" sii . .s<|. 939073 2d H. -angle oi- 50"'43« sill. s<i. 808607 3" 57'M8 35 51 Azii 9-9991 7 6 (the lesser) 2d A 7. niuth. 1..43' ' 3 sine COS. 9-34H 99728 Table f'oneition of the Latitude. 1, Part I., 32° and 87° 9'0'4 Lat. .sec. (above) 0-117 I" 26" pro. log. 2-099 Corr. of Lat. 1 1' Pro. log. 1230 The lat. being N., and both observations to the southward, it is the simplf ra?e . tho obs. being on the same side of the merid. and the computed interval the lesser, 1 1 ' is to b« added to 40" 12', which gives Lat. 40° 23' N. Ex. 2. (Different sides of Mer.) Oct. i6th, 1878, lat. by ace. 41" 22' S.. long. 150" E., at about 10*' 45"' a.m. obs. alt. Q 53° 2' 20", bearing by compass S.E. by S. j time by cliron. 6'' 29™ 19"; at 10'' 19'" 6' by same chron. obs. alt. Q 41° 1' 10', ind. corr. —3 20', height 01' eye 14 feet ; chron. ijaininy i2»-2 daily. Course S.E. by S. ; rate 6 knots. The course being exactly towiird-, tlie sun, llic run in 4'' jjivis 24 lo be uiAiwi 10 HI alt lilt pol, diats. 81° 14' snd 81" 10'; isi alt. 53° 3j'; '.'d, 41'' 9. FI.\D1\G THi: I.ATlTL'Dfc. 2G7 Alt. 51-5 5 Ut 4. 21 SfC. 0-.2465 S Dist. Si 14 cosec. OC05CS 176 «l 8S 5 COS. 8-5=434 34 30 U. H ..ngle ." .J- 34" sin. s<i 9-7?3'3 8 +o::o 53 35 3'" 972 Corre »' So ™»- 40 41 sine ngle 2i'45" 33' sin. tq. ' '3 54 3 5(^ 7 ^(A(! leisert 3 i9 47 40 i' 9 9-8*o 9-995 9938 1 of the Latitude. Table 71, Part II., 31° and 60° 9174 Lat. sec. (above) 0125 o"' 40' pro. log. 1-43' 3' pro. log. I "7 30 rhr obs. on tliff'erent sides of meridian and the computed interval the hsser, 3' lias to b» •ubtracted from 41° 22', which gives Lat. 41° 19' S. Ex. 3. (different sides of the pr. vert.) Feb. 19th, 187-8, lat. by arc. 52° 55' S., long. 1 1° E., at i" 40'" i-.M. obs. alt. Q 43° 53', bearing S.W. by S. ; at V 39"' 5' ''-M. obs. alt. Q 1 1"^ 55'. Course in int. N.E. by N., 3-5 knofts an hour ; height of eye 16 feel : required the Latitudb at id observation. 1st Alt. (run allowed for) 43° 5c', 2d Alt. iz" 3' ; Ist Pol. Dist. 78' 4"'- 2d Pol. Dist. 78' ;,'; 1st Hour-angle 1'' 38"' 46". Az. 35°; 2d Hour angle 5" 38'" 57', V. Az. 8-°; corr. of Int. 7' to be subtracted, because the obs. are on the same side of nier., the computed int. greater, obs. on different sides of pr. vert., and the hour-angle V with greater aziiiuuh. Lat. 52° 48' S. 12.] Double Altitude of a Star. 7.53. Tl'.is is the same as for the sun, except that tlic niter val by '«atel! must be increased by 10' an hour. [3.] Double Altitude of a Planet. 75-4. Find tlie Green. Date at eacli obs., ami reduce thereto the U.A. and dccl. Apply the change of R.A. to the interval, as directed No. 735, and add to t!ie interval tiie Acceleration upon it. In otlK^r respects proceed as for the sun. [4.] Dnnble Allilude rf the Moon. 7c)5. Find the Green. Date at each ohservation, and reduce thu U.A and dccl. Subtract the cliaiige of K.A. fiom tlie interval, and udil to the interval the Acceleration upon it. In other respects proce«J us I'm- the sun. 75G. For tlie Degree of Dcpendance, scc^ No. 771. 4. Ivorijs Solution, for the tame Body. ')7. Thoufjh this method apjilics, strictly, to a Wlf wliich di •lianj^c it-s dccl 'lation, yet it answers well erimi^li, in co'imi 268 NAUTICAL ASTRONOMY. practice, with the sun, by employing a mean between the pol. dists. proper to each observation. Tlie same is true of the Uioon vvlisn near her greatest declination, N. or S., since at that period she changes her decl. about 1' only in 6 hours. (1.) With the sun, the moon, or a planet, find the Green wirh Date for the middle time between the observations, and reduce thfl decl. thereto. Find the pol. dist. by means of the lat. by ace, N. or h. Correct the altitudes, and reduce them to the 2d place of observ- ation. Find the polar angle. For the sun, this is the interval in app. time; or mean time, as shewn by the watch, is near enough. For a star, see No. 734. For a planet, see No. 735. For the moon, see No. 736. Take half the interval, and find half the sum and half the difference of the altitudes. Note. — When the interval is rather small, more care is required in the work, which maj tlieu be carried to quarter minutes in Table 68, at sight. (2.) For Arc I. To the log. sine of the half interval add tha log. COS. of the decl.: the sum is the log. sine of arc 1. (3.) For Arc 2. Take the ar. conip. of the log. sine found, and add to it the log. cos. of the half sum of the alts., and the log. sine of tlu'ir half diflf'. : the sum is the log. sine of arc 2. (4.) For Arc 3. To the log. sine of the decl. add the log. sec. jf arc 1 : the sum is the log. cos. of arc 3. When the lat. and decl. are of conti'ary names, or the pol. dist. exceeds 90**, take the suppl. of this arc. (5.) For Arc 4. Add together the log. sec. of arc 1, tlie log. sine of the half sum of tlie alts., the log. cos. of their half dili'., and the log. sec. of arc 2 : the sum is the log. cos. of arc 4. [fi.) For Arc 5. This is the ditf. or sum of arcs 3 and 4.* When the observations are on different sides of the meridian ; if the pol. dist. is greater than the colat. take the diff. ; \f less, the sum. When the ol>servations are on the same side of the merid., when the pol. dist. exceeds the colat., take the diff. When the pol. dist. is i'()ii(il to or less than the colat., take out the log. sine of the lat. by ace; then add together the log. sines of the decl. and mean of the the end of tlie operation, that the computer may content himself tlie sum or dilT. gives the result in lat. nearest to the lat. by ace, as ic two results will differ greatly. A and B are the places sf the body at the two observations; PA, PB the polar distances; Z A, Z B the zen. dists. ; A P B the polar angle or inter- val. P D is drawn perp. to A B, and dividing A P B into two equal parts ; Z F is perp. to P D. Then, Arc 1 is A D ; Arc 2 is Z F ; Arc 3 is P D. As P D is usually gre-iter than A D, from which it is determined, if h small error occurs in A D, P D will be in error still more. Arc 4 is D F ; Arc is P F. P F here is PD - 1) F; but when the pol. dist. is mu.h less than PZ, F may fall beyond D on P I) prodiictd, and then P F The colat. P Z Id then iVmnd from P F and Z F. FINDINCi THE lATITUDK. 200 alts, (already eniploved). It'tliH lust sum is less than the sia. of tlie lat.,' take the dijf. ;' \{ yreatfT, I lie sum. One place in the bgs. is enough, since, if the distinction is not strongly marked, the ca-e should be rejected. (7.) For the Latitude. To the log. sec. of arc 5 add the log sec of arc 2 ; the sum is the log. cosec. of the latitude. Note.— To save reopening Table 68 at the same place, logs, ttiken out at the same op.'Q Bif(, or repeated, are marked with the same letters. Ei. 1. (Obs. «anie side.) Lat. by ace. io° S, long. 7° E. ; true alts, of the bu;i, 58 '4°'. *nd 63° o reduced to the same place ; interval, 31"' 5+': required the Latitude. R -d. of Ded. in the Form, Ex. 1. D. 2GI. Correction of Alts, in the Foim, Ex. 1, Red. Decl Pol. Dist. In,. r-- 54- Half 16 27 Decl. 14" 14' Arc 1 3 59 Half.'lum 60° 50' HalfDitr. 2 10 Arc '2 15 2Z 14° 14' N. 104 24 sin. 8-85605 9-<)S6h(«) sin. 8«4-'9 W Ar. CO. I-.578. 9-68784 (c) sin. 8-5-757 W sin. 9-423--Z W p. 2(;6, then, Ist Alt. 58° 40 2u 63 Sum .21 40 Dirt-. 4 ^o Half Sum 60" 50 HalfD.ir. 2 10 9-39566 (a) 9-3967. Arc 1 (rep.) 0-00.05 (ft) 9-94112 (c) 9-99969 (a) Arc 2 0-0.58. (e) 9-95767 (Suppl.)7 5= Arc 3 104 34' cos 26 sec ... sin 53 Ai c4 24 79 33 Arc 2, sec. (rep.) 00. 58. (<• cosec. o'757-3 Criterion for Sum or Diff. of Arcs 3 and 4 . Pol. Dist. exceeds co]at.-iliff. Lat. 10° 4' Kx. 2. {same side mer.) Lat by ace. 43° 10' N. ; alts, of Capella, reduced to the same place. 22° 58' and 56° 14'; interval'by chronometer, 3'' 34'" 17": required the Lat. Interval red. 3'' 34"' 53'; decl. 45° 50' N. ; arc 3, 40" sj'. Criterion, sin. lat. 9-8 ; sura of sines of decl. and mean alt. 9-6 ; take the diff. of arcs 3 and 4. L.\t. 43" 29' N. Ex. 3. (obs. different sides.) Lat. by ace. 10° N. ; alts, of Castor, 63° 16' and 46'-* 12' ; tnterval by a watch, 3" 55"' 25'; decl. 32° 14' N. : required the Lat. Arc 1, 24° 33i'j Arc 2, 11° 54'; Arc 3, 54° sk' i Afc 5, 78' 58'. Lat. 10° 47^' N. 7.58. (1.) When the alts, are equal, this method is peculiarly convenient. Compute arcs 1 and 3, as above. Arc 2 is 0. Vor Arc 4. Add together the log. sine of the alt. and the log. nee. of arc 1 : the sum is the log. cos. of arc 4. When the })ol. dist. exceeds the colat., the diff. of arcs 3 and 4 ia the colat. ; otherwise their sum. Ex. Equal alts. 46° 51' ; pol. dist. 66" 33 ; interval, 4'' 37" 50*. Lat. by ace. 60°. Arc 1,31° 3o|' ; Arc 3, 6i° 10}'; Arc 4, 31° 9!'. Lat. 58" 59'. (2.) When the declin. is 0, the half int. is arc 1, and arc 3 is 90°. Ex. I^t. by ace. 60" N., decl. o, int. 2*' o"' o» j true alts. 18' 53' «nd 20° 42'. Arc 1 ;°o'; Arc 2, i^^ig^'; Arc 5, 26° 34'. Lat. 59"59|'N. Note. — If the time also is required from the observation, with the outer alt., lat. found, list. (red. to time of outer alt), find the hour-angle. No The sum of log. sec. lat. and log. sld. arc 2 is log. >nij p 5J70 WACTJCAL A8TRONO:iIV. IV iU Dovm.E Altitude of Dutehent Bodies. 759. The forms of solution described in Nos. 737 and 747 for the cases of two altitudes of the same celestial body apply to the altitudes of diti'ereiit bodies, the difference of their right ascensions supplying iu ])art, or entirely, the ])lace of the measured interval. Since the value of this observation, like the former, depends upon the ditlereiice of azimuth, the two bodies may often be so selected as to aH'ord the best possible result under the circumstances, while in the case of a single body the necessary conditions are not, generally, matter of choice. Hence this method may be practised with equal convenience in all latitudes. This observation is particularly convenient in the case of two stars, because, as the right ascensions of the stars change very slowly no reference to tlie absolute time is necessary. 760. \\ hen the two observations can be obtained at nearly the same time, this method has the advantage of being independent of the rate of the watch, and also of the errors of the ship's run ; but when an interval ehii)ses between tlie observations, allowance niu«t be made both for the rate and tiie run. 1. One of the Altitudes {of Two Bodies) bebiy near the Meridian. 761. Limits. These are the same as those given in No. 745. It must be remarked, that the rules for the limits apply to the bear- ings at the time the bodies are actually observed, whether there be an interval or not. For ex., if the sun be observed S.S.E., and tlie moon E. by S., the case is a good one; but if the observation of the moon were dalayed till she bore S.E., the case would not be good. 762. 'lite Observation. Take tlie alt. of the outer body, which should be observed as nearly E. or W. as possible. Then observe the alt. of the inner one; lastly, that of the outer one again, noting the times of each alt. 763. The Computation. (1.) For the sun, moon, or a planet. Find the Green. Date, and reduce thereto the R.A. and declination; and for the moon, her hor. par. and semid. For a star. Take the R.A. and decl. from the Nautical Almanac, or from Table 63. Call the ditf. of R.A., or its supjil., the polar an(j4e. (2.) Reduce the alts, to the same instant, and correct them. (3.) With the outer alt. and pol. dist. find the outer hour-anglr, and proceed as in No. 740 (4), to the end. I about s^ 55" P.M. M.T.; Ut. ace. 40° 15' S.. long. 38'=' 52' lUo (leiluced to the a»me instant) obs. alt. .-^Idebaiaii neM i', lieiglit of eye 18 Ccet : requirfd lla- Latitude. FIXDIXG THE LATITUDE. ■Hie Gt. Datf !s 6" S'' 30". Ald.lmran's obs. all. 33° 17', true alt fcturn-s Kid. U.A. xV 34" »4' Lat 40", UecL 16' (co„tr,Ty \ Aldeluran's K.A. + 14" 2S zS 56 n„me,) J I'olHf :in);V- 4 54 32 lO" 47' sin. B,|. I'he true Alt. of Sat. 11' 41', 0" I 3 sin lat. 40" 15', |>»l. (list. 33 II 85=' 6' give Saturn's hour- Mcr. Alt. 33 »5 a„sle _i_ii_L9 Zen. Uist. 56 35 S. Aldebaran's hour-angle to 47 Uecl. 16 16 N Saturn's Uecl. 4° 54' S. pol. .list. X5'> 6' Lat. 40 J9S. Alitebaran's ilecl. 16° 16' N. Ex. 2. Feb. zd, 187S, lat. by a<c. 54° 53' \. ; obs. alt. Regnlns 15° 54', and ll .M<lebaran (reduced to tbe same instant) 51' 17'; ind. corr. -3'; I.eigl.t uf eve rejjuired tlje Latitude. iVl 7-3'3 K. A. Regulus, 10" i» 55', dccl. 12° 34' N. ; R.A. Aldebaran, 4" 28™ 57% deil. 16° 16' N. Regnlus' tiue alt. 15° ^l' : Aldcbaran's iiitto, 51'' 19'; hour angle of Kefiulus, 5'' 21'" 54", hour-angle ot Aldebain, ii'°4'; Ued. -f 4'. Lat. ^i^'^ j N. 764. When the change of alt of one of the bodies is not given b3- the ol)sei-vation, its ahitiule cannot be reduced to tlie same instant as the other by No. 660; to compute it (No. 671), tlie azimuth h required, which, if not observed with some precision, must be com- puted. But this reference to the altitude may be avoided, thus:^ Add the interval of time, increased by 1' for every 6'", to tlie II A. of the body first observed, and subtract the R.A. of the body lust observed; the rem. is the polar angle. If the sum e.xeeed 24'', reject 24'". Ex. 1st. June 24ih, 1878, lat. by ace. 40° N., long. 149° 52 AV, ; time by iliron. H' o' 1"", obs. alt. of a .\iidroined.e 41° 53', and 2" 15' al'tcrvvunls iibs. all. ul Jnpiu-i 30° 29' to the sOMtilward; height of eye 16 feet. '' 33'" >7'. R'^J- die'- 19^ 22 S., (rae alt. 41° 48'. The hour-angle of a Anrirmneil.T com. puted from alt. 41° 48', lat. 40 , and p I. dist. 61° 35', is 3'' 50'" 33". 1 he diHerenee between the polar angle and the hour-angle of o Andromed* le.ivi-s .lupiter's hour-angle 19"' 27', which gwvi Ued. + id, mer. alt. 30'' 33', and Lat. I 40° 5' N. 54° 50' N., obs. alt. Regulus 17° 21, and 3" 40' nid. corr. -5', height of eye 16 feet: required the 12° 34' N. , H A. lligel 5" 8" 42', decl. 8" 21' S. ; alt. Kegulus, 17° 9', liourangle Regulus 5'' 11" 57'; hnur- ■ 5'- Lat. 54° 59' N. 76.5. When the body nearest the meridian is observed below the pole, add the hour-angle of the other to the polar angle; the suppl. to 12" of this sum is the inner hour-angle, to which con^jnik' tiio Reduction. Ei. March imt, 1831, off Cape Horn, lat. by ace. ^(i° ^o' S., long. 65° \V., ,«t nijlit, obi tnie alt. a Pavonii 24° 38', not loni; pant the mer. below the pole; and after J" if obs. alt. y Crucis 64° 47' ; both sUrs rising, and both to tile S. of E. Red. U.A. of J ipiter 20" 33" '■ R A. ofa An< romei ;le K 0" 2" •lupiter's U..\ Polar \n V 4 33 3' Kx. 2. Jan. 3d, afterwards obs. alt. LLti.ude. .878. Kigel !a.. by 26=46 S. R.A. Regulus, p..lar angle V 56-" angle Rigel .5- 5. 10'' I 2-, t Red ° 54-. d ue alt. to this eel Re -t- 272 NAUTinAF, ASTHONOMT. y C.Tix R A. Pi.lar Angle The liour-aDsrle of y Crux, computed from alt. 64° 47', lat. 56^ ^o', and pol.ilist. 33^50', is3"6'"i8'. This hour-angle, added to the pola/ Angle, gives hour-angle ofa Pavo 1 1'' o'" S*, or 59"' 52* below the pole. The Red. 'o iiiis is 38', and the nier. alt. 24"' o' give* I Lat. 56"44' S. (Decl. of, Pavo, 57^16' S.I t. Neither of the Altitudes [of Two Bodies) being neur the Meridian. 760. Limits. These are tlie same as for No. 749. 767. Tlie Observation. Take an alt. of the outer body, then of tlie inner one, and, lastly, of tlie outer one, noting- the times. At each observation note whether the body is to the iioithward or ■southward of E. or AV. (true). 708. The Computation. The approximate method. (1.) Take out tlie right aseens. of tiie bodies from the Nautical Almanac, reducing them, if necessary, to the Green. Date. Take the ditf. of R..4., or its supjjl. to 12", for the polar angle. If the 2d ait. of the first body be lost, proceed by No. 763. The result is the polar angle. (2.) Correct the altitudes. (3.) Compute the hour-angle of each body. When the bodies are on the satue side of the meridian, take tlie diff. of the hour-angles; when on oppo.'iite sides, their sum, for the computed polar angle. If this sum, or difF., agree tolerably well with the polar angle, tlie lat. by ace. is near enough; if not, proceed as in No. 752(5) to find the corr. of lat. £.1. 1. Feb. 25th, 1830. H.M.S. Eden, lat. by ace. ii°45'S., long. 19° W., took alts, of Canopus and Sirius as following, butli stars to the E. of the iner., and both to the BOUtbward of the £. point. Canopus. 5" 43"'"' 46°58''4 5 45 ^5 47 T1. 5 4+ "8 47 2-8 Sirius R.A. 6*' 37'"4o* Canopus 6 zo 11 Polar Angle 17 jg S" 4S-0 5 5° ° 5 49 Decl. i6''29'-7 S. 52 36-58. Canopus. 5" S'"' 4' +7°^7'-4 5 54 ° 47 33-4 5 5^ 3^ 47 30 '4 Pol. Dist. 73'3o'-3 Heaucing the alt. of Canopus to the time s"" 49" gives alt. required, 47" iS -4. The . of Canopus, 47° ii'b, and of Siriu.s, 71' 56'-7. Hour-angle of Canopus Hour-angle \^ 1™ 57' Pol. Disl. 37-23' UX. 47 14 Hour-angle of Sirius Ditto Canopus Dili'. 01 comput. Pol. Angle' Pol. Anfcle FINniNG THE LATITUDR. 'J 73 T»ble 71, I'art I., 14° and 75' 9'39« L:it. sec. 0-009 «■" 34* I'l-- l"g- r^zi (,'orr. ol lilt. 34' pr. log. o'7z3 The ob«. are tn the same side of the merid. and of the pr. vert. ; both hour-ai.fslei are n oe marked V; the oomput. int. the lesser: the greater hour-angle is with tlic grfator uimuth ; 34' is to be siiblracted from 1 1° 45', which gives the l^at. 11° 11' S. Ki. 2. (The Ex. No. 765.) Tlie oomputed hour-angle of > Pavo is 11'' 5"' &•; the (lift ^< which, and 3'' 6°' 18", is 7'' 58"' 42", tlie computed polar angle, which is greater Ihao )• 53'" 50". The error is 4'" 5;'. The azim. of « Pavo is 8", that of y Crux 71^°; the coir, of lat. by Table 71, >'art I., i» e', which, since in this case the (greater hour-angle 1 1'' 5"' o' is with the lesier azin.uth, is t( ho subtracted from 56' 50', and gives Lat. 56° 44' S., as by the other solution. Ex. 3. Dec. ist, 1878, lat. by ace. 41° i8' N. ; obs. alt. of Markab, S9" 2'. and that of Allair, reduced to the same instant, 23° 38' ; both bodies to the S. and E. ; ind. corr. —2'; height of eye 16 feet: required the Latitude. R.A.. Markab, 21'' 58" 45', decl. 14° 33' N ; R.A. Altair, 19" 44'" 52*, decl. 8° 33' N. ; true alt. of Markab, 58^ 55'i that of Altair, 23° 30' ; polar angle, 3'' 13"' 52"; Mar';ub's hour-angle, i'' 11'" 44'; Altair's hour-angle, 4" 24"' 26'. Then 4'' 24'" .6'— 1'' 11'" 44" — 3'' 12'" 42". Azimuth of Markab, 35°; azimuth of Altair, 80°. Corr. of lat. 11' to be added to 41° 28'. LATiTunii, 41° 39' N. Ex. 4. May ist, 1878, lat. by ace. 29° 48' S ; obs. alt. of Altair, 26° 24', and the obs. all. of Arcturus, reduced to the same instant, 32° 23'; the bodies on dilferent sides of tlia meridian, and to the north ; ind. corr. + 2' ; height of eye 14 feet : required the Latitude. K.k. of Altair, 19'' 44'" 52*, decl. S°33' N.; R.A.of Arcturus, 14'' 10" 9" ; decl. i9°4y'N.( polar angle, 5''34"'4i*; true alt. of Altair, 26° 20' ; do. of Arcturus, 32° 20' ; hour angle of Altair, j"" 31'" 43*; Arcturus' hour-angle, 2'' 2" 3"; error, o" 56*; azimuths, 62 and 34") oirr. of lat. 6' to tub. from 29" 't'. Latitude, 29^ 42' S- 769. The error of ttie correction of lat. is directly proportional to tlie error of the interval : hence, when tlie moon is eniployeii, licr U.A. should be computed for the actual time at Greeinvich, as given liy ti.e chronometer, or found from observation of a lunar distance rather than by means of the erroneous long, by account. Ex. April 7th, 1831, lat. by ace. 34° 40',?., long. 42° W. ; true alt. J 38° 27' to the N.W. At the same time, true alt. © 47" 44' to the NE-d ; Gr. M.T. by lunar obsei-vation, jh i^cii ,j. . required the Latitude. © R.A. I*" 2'" 41', pol. dist. 96° 42'; J R.A. 20'' 52"" 28", pol. dist. 74° 10'; ®'s hour- angle o'' 36"' 45» E. ; J ditto, 3'' 3 5"' 27* W. ; © 's az. 14° ; J ditto, 81°; suppl. of dilf. of R..\. 4'' 10™ 13'. The error of the computed polar angle is 1'" 59', corr. of lat. -h6', and Lat. 34^ 46' S. This Ex. may be worked by No. 763 (3), thus : the J 's hour-angle, 3'' 35"' 27', sub- tracted from 4'' 10'" 13', gives tbe ©'s hour-angle 34°" 46'. The Reduction to this is 49', and Lat. 34'' 45' S. 3. 77/c General Solution, for the same, or different. Bodies* 770. (1.) Find tiio polar angle. This, for the sun, is properly an interval of A.T ; but mean time is near enough. I'or a star, see No. 7o;i. For the moon or a planet, see Nos. 754, 755. • Though this method is general, yet it is not well adapted to ca.ses of short interval* (No. 727) ; because, in such cases, a small arithmetical inaccuracy ni the process may produce a considerable error in the resulting latitude, as the reader may easily convince biiiiself by working examples. This is the chief ground on which an approximate and bdirect method is of^en superior, in practice, to the rigorous method. In the figure in the note, p. 2«i», omitting the lines 1' I), Z fJ, and Z K, arc A i^ A H , I tod B <>re the placet of the same body at difTcrenl timeu, or uf dilftrcnt bidii> ; aiigit B tu NAUTICAL ASTRONOMY. For different bodies, it is the diff. of their R.A. Find the polar distances at each oijservation ; in assigning these, one pole must necessarily be assumed as the elevated pole, whether the lat. be approximately known or not. Correct the altitudes, and reduce them to the second place of observation, and find the zenith distances. (2.) P'or the Arc A. Take the suppl. of the polar angle ; and add the pol. dists. together. Add together the log. sine square of the biippl. and the log. sines of the pol. dists. ; the sum (rejcctiug tens) is the log. sine square of an arc x. Put j; unuer tlie sum of the pol. dists.; take the sura and diff. and half the sum and half the ditf. Add together the log. sines of the last two terms: the sum (I'ejecting tens) is tiie log. sine square of an arij A. (3.) For the angle B. Add together the arc A and the two polar dists.; take half the sum, and from it subtract the arc A and the outer pol. dist., notmg the two remainders. If the half sum is the lesser, subtract it from the other quantity. Add together tiie log. cosec. of A, the log. cosec. of the outer pol. dist., and the log. sines of the remainders: the sum (rejecting tens) is the log. sine square of the angle B. (4.) For the angle C. Add together the arc A and the two senith dists., and from half the sum subtract A and the outer zen. dist. ; note the two remainders. If the lialf sum is the lesser, sub- tract it from the other quantity. Add together the log. cosec. of A, the log cosec. of the outer zen. dist., and the log. sines of tiie two remainders : the sum (rejecting tens) is the log. sine square of the angle C. (5.) For the angle D. This is the sum, or diff., of B and C, according to the following directions : — In the case of the same body. I Pol Dist. greater thar Colat. Dist. less than Colat. greater Alt. greater Alt. ith tenner with greater Pol. Dist. greater than Colat. Pol. Dist. less tlian Colat. Interval diff. Note. — The difference of bearing in the interval must be less than i8o°. Is P B A ; angle C is Z B A ; angle D is P B Z larger and P A smaller, P B Z may be P B A + Z the included angle PBZ, give PZ. In the case of two stars, A and B ayi very ' computed for certain pairs of stars, and inserte Diatetlally shortened. — Tal/les for facilitating i LmnT. Shadwi.i,l, R.N. 1836. which is PBA-ZBA. When PZ is I A. Then the two sides PB, BZ, witD early constant, and have accordingly b<^en . in tables, by which the L-oni|Mitiition is ic Cowjiittatioit of Double Alttltidet, by FlNniN'n THR I,ATITTinF.. 27: 'd.) For the Latitutle. Take the siinplemeiit of D to 180" Take tLe sum of the outer polar and zenith distances. Add togetlier the loj^. sine square of the suppl. of D and the log. sines of the outer pol. and /.en. dists. : tlie sum (rejecting tens) is the log. sine square of an auxiliary arc y. Put tliis arc under tiie sum of the zen. and pol. dists. ; take the sum and diff., and half sum and half dilf. Add together the log. sines of the last two terms: the sum (re- jVoting tens) is tlie log. sine square of the colatitude, reckoned from tlie same pole as the pol. dists. Ex. I. Interval. 31"' 54'; the 1st and outer alt., corrected and reduced to the 2il place, ll ^8' 39'42"; the 2d alt. 62^ 59' 36" ; outer pol. dist. 104° 24' 30" ; the other, 104 '>4' i»". For the Ar A. Interval 0'' 3^"' H* Suppl. „ 27 6 sin. sq. 9-997761 Pol. Dist. 104" 24' 30" sin. 9-986.2, Pol. Dist. 104 24 12 ain. 9-986130 Sum 20X 48 42 Auily. arc x MO 3 4^ sin. sq. 9-970012 Sum 358 5^ =4 Diff. 5« 45 Half Sum 179 26 12 sin. 7992640 Half Uiff. 29 22 30 sin. 9-690660 Arc A r 57' 5^" sin. sq. 7-683300 For the Angle B. For the Angle C. Arc A 7° 57' 52" cosec. 0-858367 A re A 7° 57' 5^" "'««• 0-858,67 Ower p. d. 104 14 30 cosec. 0-013879 0<tterz d. 3. zo ,8 «,sec. 0-283CX. Iiinrc p J. 104 24 12 I inerz d. 27 24 2,6 46 34 66 ,8 34 108 23 17 33 9 '7 100 25 25 sin. 9-992773 25 ,1 25 sin. 9-62902* 3 5« 47 sin. S-84.384 I 4« 59 si"- 8^^oH Angle B 90° 59' 20" sin. sq 9-706403 Angle C 5 1° 16' 3 1" sin. sq 9-'-7»33o The observations are on the sa ne side of the n eridian, and the pol. dist. oTeater tliau Uin colut. : beuce D is the diff. of B a id C, and is therefore For the Latitude. 39'' 42' 49".* Arc D 39° 41' 49' Suppl. 140 17 II sin. sq. 9-946759 Outer Pol. Dis . 104 24 30 sin. 9-986,. I Outer Zen. Uis t. 3' ^0 .8 135 44 48 sin. 9-716079 Auxly. Arc y 83 45 =0 2.9 :o i 51 59 28 J09 45 4 sin. sq. 9_;64895'' sin. 9-973668 sin. 9-64.773 »5 59 44 79" '■S' 24" an. sq. 9615441 Latitudr 10 4 T^s. • A general rule for assij^ing the sura or the diff. of B and C, in the case of <itj'<;*ni 276 NAUTICAL ASTRONOMY. This proress is less troublesome than it appears. The 1st ami 4th steps are of the same fono. as ate, also, the 2<1 und 3d.* Ex. 2. Lat. by ace. iz° S. ; true alt. of Sirius, ■jj°$6'^i', pol. Jist. 73° 50' j8"; true t'A. of Oanopus, 47° 13' 36", pol. dist. 37° 23' 30" j diff. of R.A. 17"' 29'. Both stars to the eastward, and Sirius the outer one or easternmost. The arc x is 99° 22' 15" ; A is 36^ i6'45" ; angle B, 4° 30' 10" ; angle C, 100° 10' 33" : the angle D, the sum of B and C, is i04.°^o'43'. The arc y is 38° 54' 38", and the L.\r. 11° 13' 27" S. 771. Degree of Dependance. The lat. by double altitude is affected by the errors of altitudes, pol. dists., and interval, or polar angle. The effect is the same, whether by the approximate or rijj-orous process. (1.) To find the error of lat. caused by 1' error in one of the alts. To the log. 3-431 add the log. sine of the azimuth at that alt. and the log. from Table 71 : the sum (rejecting tens) is the prop. log. of the error rcq;i;red, nearly. Ex. Suppose in E-x. 1, No. 768, the alt. of Canopus i 3'41 9-3S 9-39 The Error of Lat. is tlif fore about 3' 24". (2.) The error of pol. dist. will be worth notice only in the case of the moon, in consequence of her rapid change of declination, and the uncertainty of the Green. Date. Find the error of each hour-angle in which the moon's pol. dist. is involved by No. 615 (3). This gives the error of the computed interval; and the error of the correction of lat. is the same part of tlie corr. itself, that the error of the computed interval is of that interval. (3.) The error of the rate of the watch will rarely be sensible. todies, would require the hour-angles to be known ; but the obsener who is well acquainted with the positions of the circles, as shewn in the figures, p. 162, will perceive at the time o( obseiTation how the angle D is composed. * When the lat. is found, the hour-angle and azimuth may be computed thus : — For the hour-angle. To the log. sine of D add the log. sine of the outer zen. dist. (already taken out) and the log. sec. of the lat. : the sum is the log. sine of the hour-angle corre- sponding, or of its suppl. Circumstances will usually decide ; but, in a doubtful case, take the sum of the log. sines of the decl. and lat.: if this is less than the log. cos. of the zen. dist., the hour-angle is found ; if greater, take the supplement. For the az'.tKuth. To the log. sine of D add the log. sine of the outer pol. dist. (alreadv taken out) and the log. sec. of the lat. : the sum is the log. sine of the azim. or its 6Uf pi. If this is doubtful, when the sum of the log. sine of the lat. and cos. of the zen. dist. is less than the log. sine of the decl., the azim. is found; if greatei, take the iuppl. Reckon the iziit iti from the N. in N. lat , and S. in S lat. jrrNlUNG THE LATITUT,R, 577 V. Ry THF. Altitude of the Pole St.4k. 7V2. 'Hie Obsercation. Observe tlie alt. of tlie pole star, noting trie time. On shore, note also the thermometer and barometer. 773. The Computation. At Sea. (1.) The error of the W.itrh on A.T. being known, take the R.A. of the sun from the Nautical Almanac, or Table 61, and add the A.T, of observation to it: the result is the R.A. of the meridian. (2.) Correct the alt. for index-error, di]i, and refraction. (3.) Enter Table 61 with the R.A. of the nicr. and the alt. ; take out, the correction, and apply it as there directed : the result is th° latitude, north. Ex. 1. July 5th, i%<)0 at ll" l" P.M. »l>p. time, ohs. alt. of the \> lestar, si°2o'; iii.l. corr. + 2' ; height of eye 16 feet: re- quired the Latitude. App. Time ii*' J"' R.A. © 6 58 R.A. Mer. li * Obs. Alt. 51° 20' li,d. Corr. +2'1 Table 38 - 5 ) "" ' 51 J7 Igi- o">. Alt. 50' Lat. IT'S N. I XcyO, at j' ired the Latitude. App. Time R.A. © It IT 38 56 R.A. Mer. * Obs. Alt. 53" 5-' liiH. Corr. -3 1 Tabic 38 -4 ) — 7 .5^on..a„. 5^° Lat. 53 44 54 53 774. Accuratehf. (1.) Find the Greenwich Date; reduce to it -111' Sid. 'I', at mean noon ; take oiit the star's R.A. and decl. from die Nautical Almanac, and find the pol. dist. Find the star's hour-angle. (2.) Correct the altitude, accurately, (3.") For the 1st Correction. To the log. sec. of the hour-angla add the jrop. log. of the pol. dist.: the sum (rejecting tens) is the prop. log. of the 1st Correction. For the 2d Correction. To the log. cosec. of the fiour-angle add tiic prop. log. of the pol. dist ; double the sum; add to this the iMnst. 1-5821 and the log. cot. of the altitude: the sum (rejecting t»;ns) is the |)rop. log. of the 2d Correction. (4.) When the hour-angle is greater than G*" and less than IS*", oaM tlie 1st Corr. to the altitude; when the hour-angle is less than 8'' or prealer than 18'', subtract it. Add the 2c' Correction in all cases 278 NAUTICAL ASTRONOMY. Kx. July 24tli, 1S90, long, o'' e™ W. ; at lo '24 " 12" 8 ohs. a t. of I'olarli. in quicksilver, 109° 36' 40"; ind. curr. -i' 30", ihe rm. 62^, bar. 30 melics: reijuircd Gr. Date, 24tli, 10" 30-" 13" Sid. T. mean noon, 2411. S" S'" 1S-2 10'' i'"3S'6) 30- 4 9f- +1 43-5 13' oj *n.A. ,h^^„2-l^ lied. Sid. Time S 10 17 Dccl. M.T. 10 24 12 8 Alt. 109 '36' 40 K.A. Mer. iS 34 14 5 Ind. Curr - 1 30 »R.A. -I 14 21 2)109 35 'o llour-an^zle 17 20 12-4 54 47 35 Or ' 5 20 f2 4 1st Coir. 2il Corn 1; T :. ":■} -40 S" 20'" 12' Soc. 07625 cosec. 00066 ■J'rue Alt. 54 46 55 1M> l°l6'56"P.L. 03092 r.L. 03692 1st Coir. 13 '7 IsiCorr. 13' I7"P.L. 1 1317 03758 2d Corr. Lat. I 1 1 2 S3 • 23 075 r6 Const. 1-5821 54" 47' cut. 9S4S7 2d Corr. l' ii"P.L. 2 1824 775. Degree of Dependance. The error is very nearly the same as that of the alt., as a small error of time produces but little efifect. N.B. — The Nautical Almanac method for obtaining Latitude from Pole Star is strongly recommended. Every year tables are calculated expressly for this purpose. Where accuracy is re- quired, as in observations for latitude made on shore, these yearly tables should always be used. CHAPTER VI. FlXDING THE TiMI: I. By a Single Altitude. II. By Difference of Altitude NEAR THE Meridian. III. By Equal Altitudes, IV. Eating THE Chronometer. 776. In consequence of the perpetual revolution of the celestial bodies, the hour-angle of any one of them affords the measure 0/ tiuie, No. 471, &c. By whatever method, therefore, the hour-angle may be determined, the time may be deduced. At sea, where the only fixed object to which the ever-ch;inging positions of the celestial bodies can be referred is the horizon, altitude is the only means of determining the time. I. Bv A Single Altitude. 777. The sun's hour-angle being app;irent time, when his iilt. is observed, the time is at once determined. In the case uf iiny viIum UNDINO TlIK TI.HR. 2 ,-<) celestial body wliicli does not pass tlie meridian witii the sun, it is necessary to allow for the difference of tiieir iiour-aiigles, or of their right ascensions (No. 471), at the instant of observation, by referring both bodies to the first point of Aries (from wliicli R.A.is reckoned), us will be described. 1 . A Ititmk ahuve the Horizon. 778. Limits. The body should be nearly E. or W., because, when on the prime vertical, errors, botii of the latitude of the observer, and of the altitude observed, [)roduce liie least effect on the iiour-angle. In genei-al, however, the body niay l)e observed at any time, wliile moving at tlie rate of not less than (i' of alt. in I'" of time; because in this case an error of 1' in the alt. will cause not more than 10' error of time, and the same error of hit. will in the same case cause a still smaller error of time. Tiie smallest azimuth, reckoned either from N. or S., which tlie body can have under this last condition, is seen in Table 46, in the column off)'. On the other hand, the alt. should not be observed when small, a>, for ex., under 10® or 15", on account of the uncertainty of refrac- tion, especially in very hot or very cold weather. 779. In lat. 60° 24' and upwards, 1' error of alt. niust always cause more than W error of time; the body should therefore be observed as nearly E. and W. as possible. In the tropics, on the other hand, tlie time may often be more correctly deterndned, when the body is less than an hour from tlie meridian, than at several hours from it in high latitudes. At sea, the uncertainty of the sea-horizon may sometimes be removed by observing to opposite points. Errors of alt. proper to the instrument, or to the eye, are obviated by observing the alt., of the same measure, on opposite sides of the meridian. [1.] Tnfnd Apparent Tme, and thence Mean Time, by the Altitttde of the Sun. 780. TheOktervfition. Observe a set of altitudes, (Number 557) at the projjcr limits, noting the times. See also No. 535. For accuracy, note the thermometer and barometer. 781. The. Computation. {] .) Having found the time corrc- upondiug to the altitude, find the Cireen. Date by the chronometer No. 575, which will be mean time; or by the time roughly estimated and the long, by ace. No. 576, which will generally be App. Time Hediice to this the sun's declination. No. 580, or, i'or common ]>uv- poses at sea, this may be done by No. 579. Find tlie sun's polar distance. No. 443. VA'hen mean time is rerpured, reduce the Equation of Time No. .583 or 584. (2.) Correct the alt. at sea by No. 647, or, ifgreatei- accuracy '\* re<piired, by No. 649. (3.) Compute the sun's hour-angle. No. 614. (4.) Mian the sun is to the M'. 'or p.m.), this liour-anglc is 280 NAUTICAL ASTROXOMT. Apparent Time; when he is to the E. (or a.m), subtract the hour- augle from 24'' : the remainder is A T. reckoned on the day before. (5) For Mean Time. Apply the reduced equation of time as directed in p. I. of the Nautical Almanac, or in Table 62, to the App. Time : the result is Mean Time. The difference between the time of observation, as shewn by the watch, and either of these times, is the error of the watch on that time. Ex. 1* Jan. I2th, 1902, at .sea, at about 9'' 30"" a.m. app. time; lat. 35° 35' N. ; lonj;. 14° W. : height of eye, 30 feet ; ind. corr. +4' 30''; ol>.s. alt. of .sun as hel"iw : rei|uiredjpp. and mean time, and the error of the watch on eaeli time, at the instant of ohserva|Ki. Note The differenres of the alts, and tlic timis are tak( means of their agreement with eacli oilier, No. 556 Dirt'. o" 35' 3i 34 l\ 32 7 ^^ _3^_J4 ■5 7 46 Time 9 31 33 Times by W. 9" 30"' 28- 27 Long. 14° W G.A.T. Jan. 22 56 26 D<c Corr . 11" Decl. nist. 21 -^8^ 19' 52 s. Ked 2!" 45 27 s. Tol. FiT 45 I7 K.|. C. ri lie.l lime Kq.l I'' 8 13 Alt. lat. P.D 22° 35 III 39' 55 45 — c c 170" 62 19 9 30 S 5 A.T gle2' 21 27" 32 45 * n. sr .i 5S 0-03207 8-92710 9 94793 aken to test their accuracy b Alt 22° IS' 20" ',''":„ 2? 4 '*° 132 so Alt 22 26 34 Os. Alt. 22° 26' 34" Inde X error + 4 30 Tahl e 38 +80 r,ue Alt. 22 39 4 A.T. at Ship 21- 32- 45" Watch 21 3' 33 \V;,tth low for A.T. I 12 AT. at Ship 21 32 4S E<1. Time ^ S 13 :\I Ti- le 21 40 58 Wat.h 21 3> 33 Slow f, r M.T. 9 25 Clirnn, ol ( ) ™ior;'°"39-49- Chr. f, ton G :M.T. -2 31 G.M.I ofObs. 10 37 18 Ship M .T.ofObs. 9 40 5« 56 20 Long. See No. 8'27 14° 5' o" W. Ex. 2. JLirch 12th, at about 4'' \$"' p.m. mean time, lat. 50° 48' N., liing. 65° 5S' E. ; obs. alt. ij_ 14° 50' 10" ; curvesponding time by W. 4I' 13'" 54-; ind -2' 20" ; height of eye, iS teet : re.iuired A.T. and .ALT. and ' of the watch G.iM.T. :Maieli II'' 23- 51'", pol, dist. 93° 15', tme ait 14° 55', Eq. T. + 9"' 55" ; ho ancle i- m or A.T. 4'' 5'" 54'; walcli t'mt un A.T. 8'" ; IM.T. 4'' 15'" 49', walcli stow m'.T. i- 55-. • In thi.s exanip'e some of the quantities arc noted but at sea the nearest minute (to which the hour-angle i unless the observuliou itself is remarkably (jood. lUcdj is generally enough, FINOIXG THE TIMK. 281 T.x. 3. Oct. zotli, 1S7S, at sea, at 4'' 40" p.m. app. tii h.'i'iht i)f eye 16 IVet ; iiul. corr. - 2' ; at 4'' 28"' 56" b) w: A. r. and M.T. and llie Error of the Watch on each. pol. dis'. 79° 31', 1 A.T. S" 4O ; .M.I true alt. 23° 15', Kq. 4>- 17- 3f; Waich/a [2.] To find Mean T,mt. the Altilmhofa 782. T)it'. Ohservation is the same as for the sun, Nos. 541, 542. 783. The Computation. (1 ) Having found the means of the times and the altitudes, take from the Nautical Almanac, or Table 6:'., the star's R.A. and declin., and also from the Nautical Almanac, or Table 61, the sidereal time at mean noon for the given day. (2 ) Correct the altitude, No. 652 or 653. (3 ) Compute the star's hour-angle, No. 614. (4 ) When the star is to the W. of the meridian, add the hour- angle to the star's R A. ; when to the E,, subtract the star's hour- angle from its R.A. (increased if necessary by 24'') ; the result is the R.A. of the meridian. From the latter (increased if necessary by 24'') subtract the sidereal time at mean noon ; the rem. is the approximate M.T. P>om this last subtract the Retardation upon it, Table 24. Take out the Acceleration for the long. ; in W. long subtract the Accel, from the result, in E. long, add it ; the result, if less than 12'', is Mean Time; if greater than 12'', reckon the time on the preceding day. (5.) For App. Time. By the M.T. obtained, and the long, by ace, or by the chronometer, find the Gr. Date ; reduce the equation of time and apply it as directed in p. II. of the Nautical Almanac, or the contrary way to that directed in Table 62. Ex. 1. .Imii. 1st, 1902, P.M., lat. 50' 46' N., lonf;. 6l° 37' W., at 7'' 56'" iS' I)y watch olts. alt of Procyoti 15^ 40' to the S. ami E., eye 20 t'ect, ind. err. o' ; reiiuired the Muui and .ipp. Times, and tlie Error of tlie Watch. Procyon's R..-V. 7' 34" lO" j Decl. 5° 28' N. ; Sid. T. mean noon, 18'' 40" 48". Ind Corr. o' Tahlc-JS -8 15° 40'. Alt. 15° 32' _o I I-at. 50 46 True Alt. 15 32 Cl.r.last.mGr. ^215 (Jr. MT. 12 9 1 1 Ship M.T. _8_3 li Long, in Time 464 Long. 6l°3l'0"W. I lie Ued. Kq T. is 3" 34' 111* wulch sluw on X.'V. 3' U)\ P.D. 84 32 150 50 75 25 59 53 I Hour-angle -4I' 48- 12* c. 0-19895 * U.A. 7 34 lo isec.o 00198 R.A. Mer.( + 24'') 245 58 I Sid. T. M. Noon - iS 40 48 s. 9-401031 Approx. M.T. 8 5 10 ic 9-93702 1 Hit. - I 1 9 '•*i1-9'5j*>98 ~8 3~5f Accel. 61' 37' W. ^ ^40 j M.T. 8 3 TV I Time hy Watch 7 56 18 I Watch >hw on ALT. 6 53 ch tublraclcit from M.T. gives A.T. 7'' 59'" 37-, and 2.S2 NAUTICAL ASTKONOMY. Kx. 2. April 27tli, 1902, A.M.. lal. 20° 47' 45" S , loiij;. 31° f E. at 2'' 19'" 41" liy tell, <.biaiia-d true alt. ul Altair 25° 14' 20' to the E. ai.d N. : iKiinnd tlie M.l. o( U.A. 19'' 46'" 2', Decl. S" 36' 35" N., Sid, T. M. Noon 2'' lis'" 9". Alt. 25° 14' 20" Hour-angle -3'' 37" S* J HI. 29 47 45 s ec. 0-061561 • U..\. 19 46 2 I'.D. 98 3ft 35 ' osec. 0-004920 I! .V .Mer. 16 S 54 iJ3___38 40 ,Sid. \. M Noon -2 18 9 76 49 20 c 's. 9-357794 Approx. :\1.T. I3"50 45 51 35 s II 9-.S94046 K.t. 16 3^ 37- S« 11 scj. 9 31S321 "U 48 29 Accel. long. 31=7 E. t-o 20 Mea.n Ti.< E '"i3^S^ 49 784. The Observation is tlie same as for the sun. Sec, also, Nos. 540, 541, 542. 785. T/ie Coinputation. (1.) Ilavino; foiiiid the means of tlie times and of the altitudes, Kiid the Gr. Date as nearly as possible l)y the chron., No. 575, or 1)V the estimated M.T. and long, hy ace, No. 576. Redtice the moon's R.A., No 591, and dccl.,' No. 589, and thence her pol. dist. ; also her horiz. parall.. No. 586 or 587, and semid., Table 39. (2.) Deduce the ajip. alt., No. 654. Take ont the correction of alt.. Table 39. Correct the altitude. (3.) Comjtute the honr-anglc, and proceed as for a star, 783 (4). F.x. 1. .luly list. iS 8. A.M.h.t. 39' 57' v., loMfT. 8° 53 E.; ALT. at Green, by cli >' ll" 4S'", ohs. alt. 2) 74° to'E. ofnter.; eye 16 feet. >\ U.A. ("orr. o'' 33-19' 1 26 Ohs. Alt. Uip. -4'1 24° lO Red. It. A. 34 45 Semid. +.5/ + II J's Red. H.I 54' 13" 24 21 j-s Aug. Sem d. '4 53 Corr. Par. + 47 3's Decl. 8 '26' 35" N. True AU, 25 8 Red. Decl. 8 27 37 N. 90 ys R.A.(f 24'') o''34-4s' liour-angle -4 "6 45 I'ol. Dist. Si 32 23 Ii..\. or mer. 20 18 Alt. 25^ 8' .Sid. T..M. Noon 7 52 32 l.at. 39 57 sec. 0- II 543 Approx. IM.T. at sliii 12 25 -.8 IVI Dist. Si 32 coscc. 0-00476 Ret ' 146 37 12 "2YT6 73 i?i COS. 9-45822 A,-eel. for 8° 53' E. -6 48 .oi sin. 987226 M.T. at Slnp 12 23 31 4" le^+s- sio. sq. 945067 id, 187S, at about <)^ 30'" p.m., 1; ofn.er., tiineby watch g'' 24"' 27- 42" 40 Mars 23" 43' of Watch. O. r. Feb. 12" 18" JO", Mar's Red. li.A. !*■ 47" 13-, Red. Oeel. i <> 10 N., T.iie Alt •r 37'. 2S3 Ilm.rangle 4' 53" 39- W. Mais' R.A. J 47 33 It. A. of Mlt. 7 4' ■» Sia. T.M.Noon 22 9 I Approx. M.T. 9 32 '° Itet. -I 34 9-543> + r'<565' 9-55i97 7Sf). When the true G.M.T. is given hy a clironoineter, tlio melon's R.A. and decHnation may be correctly found. Wiien the moon is at her greatest declination, N. or S., a small error in tiie (n; Date will but slightly affect her pol. dist. An error of 1™ in the Gr. Date causes about 2^ error in the moon's reduced R.A. 787. If the errors of the watch, as found by observation of two bodies on different sides of the meridian, but on the same side of the prime vertical, by the same observer with the same instrument, bu not identical, that error is nearest to the true error of the watch which accomi)anies the greater or outer azimuth. If the azimuths are equal, the mean of the errors is tiie true error. 788. Deciree of JJepeiulance. The alt. and the hit. being in general, at sea, more or less uncertain, and the pol. dist. of the sini and moon being reducible with precision in certain cases only, the time is in general liable to three causes of error. See No. 615. When k is jiroposed to test the observation, the parts to 30'' for the sec, &c., will be taken out with those c[u:uitities. 2. /j// tlii; Altitude 0, or fJie ImhIi/ on the Ilviizon. 789. In low latitudes the entire orb of the sun is, during certarn seasons, frequently seen at rising and setting; and in the variaiile climates of higii latitudes it is occasionally visible, though more usually clouded at tlio.se times. When the instant at which either limi) touches the horizon can be distinctly noted, the time may be determined approximately; and though the degree of approximation be rude as compared with some other methods, yet the result may often be valuable, especially after one or more days without obser- vation. It is also a recommendation to this method, as a resourci; when others fail, that it is indejiendent of Q\QYy instrument except a watcrli or other means of measuring time.* (1.) Find the time of sunrise or sunset in Table 26. Apply to this the long, in time, as directed, No. 576 : the result is the Given. Date. Reduce ilie declination, and find the pol. dist. (2.) To the horizontal refraction, 33', add the depression. Table 8, and from the sum mhlract the semid. when the lower limb is olw ibf resulls with thoie 284 NAUTICAL ASTRONOMY. served, or (uld it when the upper limb is observed : the result ia the angular depression of the sun's centre below the horizon at tl'e instant of observation. (3.) Compute the iiour-angle of tiie sun below the horizon by No. 642, using, instead of 18°, the sun's depression.* (4.) At sunset tiiis hour-angle is app. time; at sunrise tai<e tiie su]ipl. to 12 Iiours. Kx I. May ii>li, 1S78. l.it. 51° 20' N., lonjr. 26' W., obscrvc.l the sun's lower limb at SctliiiK iKuili the lioriziin at 7'' 40™ 56" hy watrh ; eve 16 feet; required App. Time. p Deel. 18°, Table26^n«es App. Time Sun.vet 7" 35", L..ng. 26° W. i^ 44 (i.A.T. I2th ^ 19 Dec), nth iS°io'N Corr. +_6 «ed. Decl. 18 16 N. Hor. Refr. 33' Depr. o'2i' Depr. 4 Lnt. 51 20 sec. 0-2042-/ 37 P.D. 71 44 eosec. 0-02246 Semid. -16 123 25 Depr. Centre 21 . 6. 42 sin. 904472 61 21 COS. 96S075 A.T. 7'' 40" 7- sin.sq. 9S5210 Watch _7 4° 56 49 Watch /i.^ Kx. •_'. ,ear,-,i 01 r.r. Da I4i!i, 1S78, lat. 18° 39' N, lonij. 62° 3c' E., the snn's upper limb at rising lorizon at 5'' 46"" II" b_v watch; eye 20 feet: required App. Time. ,jrt ,^^1, om_ red. decl. 8° 3' S. ; depr. of sun's centre 54'; Hour-angk 5'' 52'" 54'; App. Time 6'' 7'" 6' a.m.; watch 2o'» 55' slow on A.T. 790. Degree of Dependance. This we have at present no certain (latxi for determining, more especially when the observation is taken from a considerable elevation, as from a hill. The terrestrial refraction does not, it should seem, affect the instant of the apparent passage of a celestial body over the visible horizon, since the rays of light from the horizon and those from the body are similarly affected ; and hence the uncertainty of the result is probably due entirely to that of the astronomical refraction at the time and place. It may be proper, accordingly, to admit an error of 2', at least, in the refraction ; and the effect on tlie result is then fbinid by merely adding together the parts for 30" of the cosine and sine, dividing the sum by the parts for P of the sine square, and doubline the result. Ex. wliich, ( In Ex. I, above, the parts an mhled, is 15", ilie eHect due \ nd 116: tl nor in the • In the tropics the method No. 638 may be substituted, using log. sine depr. ® cent. As an aid to the working of a snn cbronometer, Davis's " Chronometer" Tables will be found very useful: they contain hour angles calculated exactly for degrees of Laiitudh, Altitude, and Declination, with means of making allowance for the minutes w''ich must Ixi taken into account. J. D. Potter, 145 Minories, London, E., price 10s. 6d. flKDING TIIK TIMB. 285 IL Bv Difference of Altitude near the Meridian. 791. When tlie sun is too near tbe meridian for a satisfactory observation of a single altitude, the time may be determined approx- imately, and sometimes nearly, by means of the observed difference of alt. in a measured interval. The method has been already introduced in the Short Double Altitude, p. 256, and it was on the ground that the same observation might be usefully employed for 'I'ime also, tiiat the small corrections from p. 223, which are scarcely appreciable in the resulting latitude, were applied. It is also worth while, in finding the time by this metliod, to correct for change of declination. The method (as already shewn in Case II., p. 2.59) is available with alts, taken on both sides of the meridian ; but, as this case would be comparatively rare, the rules have been arranged for observations on the same side of the meridian only.* 792. Limits. The observations should botli be within an hour from noon. The interval should constitute a large portion of the mid. time from noon ; but it should not, generally, amount to the whole time from noon. The Observation is that in No. 726. 793. The Computation. (1.) Reduce tlie declin., by the long., to noon at the place, wliich will be near "uough. (2.) Find the interval, and correct the second of the times by watch for the rate iu the interval, when considerable. Correct the alts., and reduce the Ist to the place of the 2d ; find their mean and their difference. Correct the ditf. of alts., and also the interval by tiie quantity in the Table, p. 223. f (3.) Compute the hour-angle at the middle of the interval. No. 729 (2), and add half the interval. When the observation is p.m. this is App. T., and being compared with the second time by watch, shews tiie error of the watch. >\'hen the observation is a.m., take tlie suppl. of this time to 12''. Note. If the rising ur falling of the sun lia« not hfen distinctly noticed, or it is uncertain whether the alts, are on the same or iliffereiit sides of the meridian, ascertaiu the fact by the precept, No. 72S. • For the like reason, namely, not to increase unnecessarily the number of precepta, the observation below the pole is not treated ; this presents no difficulty. t T'lis is the (|mntity which, added to the sine, ni.ikes it equal to the aic, and by me-iiiD of it re einplo) the table of «ine» equally well f(,r arcs. 28 G NAUTICAL ASTRONOMY. Ex. 1. Muy i4tli, 1878. about 1 1" a.m., lat. 48° 4' N., long. 21° 11' W., at 1 i" 28" kT by watch, ob». alt. Q 58° 9'; at ii'' 51"' 50' by watch, oLs. alt. Q 59° 39' ; ind. coiT. — 1' 20" ; height of eye, 16 feet ; rate, 5^ knots ; a-head at Ut obs. : required the Ermj of the Watch. I'imes by 1 1 1"= 28°' 10' Watch ] 11 52 ;o Interv. Corr. 24 30 ^4 3 3 ® Decl. 14th iS°io'N. 2.''W. _4_I Uej. Decl. 18 41 N. Diff. Alts. i°i7' 51' Interv. -4'" 3 3' Decl. iS 41 Mean Alt. 59 5 Hour-angle o'' 44'" 4.(.' Alt. 58° Ind. Corr. Dip Alt. Q 59° 39' o' Corr. A Semid. 59 33 40 ZJI 59 33 10 + 15 51 58 .8 59 2d Alt. 55 49 Alts. 58° 21' II ■ 59 49 ' Mean Dift-. Carr. 1st Alt. 58 21 II sine 8-4C74 , Hour-angle cosec. 0-9710 Comp. Mid. T. sec. o-iyco Half Int. *^''' '^'°-,^,ll i T. of 2d Obs. < CO,. 221^^ Do. sine 92877 I computed by Watch Watct fast 15 -6 27 31 Ex. 2. Lut. 10° 41' .S., red. de.'l. 20° 56' N , alts. © mputed App. Time of 2d Observation, o'' 39'" o». 25 19 interval 12'^ 14* 794. Correction for Change of Declination. Wlieii tlie sun is on tlie iiR'iiJiaii, his motion in declination (which then takes place on the meridian) is \ieiy. to the horizon, and consequently affects the alt, by exactly the same quantity. When, on the other hand, that part of the sun's celestial meridian or declin. circle, on which he is, is parallel to the horizon, his change of declin. does not affect the alt. at all. Hence the correspondinij change of alt. is always between and the whole amount of change of declination. The 2d alt. differs therefore by the whole, or a part, of the change of declin. in the interval, from what it would have been had the decl. remained constant. When the motion in declin. tends to increase the alt. the 2d alt. is too great; otherwise too small. There is, however, no necessity, in this method, for a very nice process of correction, for when the mer. alt. is small, and the sun not far from the meridian, the motion, in declin. corresponds very nearlv to that of alt., and the entire change may be apjilied; and when, on the other hand, the nier. alt. is great, the motion in alt. is so rapitl, that a few seconds, in the estimation, are of no consequence in practice, or the whole quantity may even be neglected. Ex.1. May 3rd, 1878. lat. 26° 14' N., long. 161° W., at lo"! 31"' 1 8" bv watch, obtained true alt. © 71° 49', and at 11'' 7'" 2i» true alt. 77^ 46': find the Error of the Watch. The Hour-angle is 46™ i8*, Mid. T. 1 1"" 13"' 42., and Watch s/ow 24"' 22'. Ex. 2. Nov. 4th, 1878, P.M., lat. 63° 46' N., long. 54° \V. alt. Q 10° iS' i", and at 2'' 36™ 27' obs. alt. Q 10" 2' 29'. 16 feet, the ship having no way. The diff. alts, i 5' 40', and Int. 21™ 32' (corr. by I'l, give of decl. 17', added to 2d alt. gives diti". alts. 15' 2 56* by watch, + 2', height of Degree of DepenJance. As the interval may be meusiuwl FINDING TIIV T(MR 287 with precision, and as tlie lat., declin., and alt , are required approx- imately only, tlie value of tlie result depeiuls aluiost entirely on tbe dirt", alts. (1.) The error of the mid. time due to a j;iveii error in the diti". alt. is founu by taking away the sine employed, juid adding tiial of the dirt', alts, vitiated by a proposed error. The residt is more ti'ustvvorthy as the diff. alts, is greater. Ill Ex 1, No. 793, lat. 48° 4'N., an error of 30" in the ililV of alts, causes ic' error o< time ; tlie obs. alts, would be better nearer noon. In Ex. 1, No. 79 1, 30" error of diff. alts, eauses 4' erioi of time. In Ex. 2, No. (S.'J, 30" error of diff. alts, causes 21" error of tiii.e. In E.\. 2, No. 794, lat. 63" 46', 30" error of difi'. alts, causes 48". The rase is unfavour- able from the smallness of the motion in alt. (•2.) The chief meiit of the method is its insensibility to an (Mi'or in the latitude, which, under the same circumstances, renders the observation of a Single Alt. useless. The effect of a proposed error is found by changing the sec. lat. before employed for the sen. of the lat. proposed. In the following examples the effect of an error of lat. in the result by Single Alt. also is uoted for conijiaiison of the two methods. lu Ex. 1, No. 794, lat. 26° 14', 10' error of lat. (that is, using 26° 24') causes only 4* error of time. The effect of this error on the time by the single alt. 71^ 49' would be 28". In Ei. 2, No. 793, 10' error of lat. causes i" error of time. The error of time bv the single alt. 57' 17' would be 2'" 9'. Since a single alt. very near the meridian cannot be employed for finding the tiuie, and since the latitude at sea is usually uncer- tain some miles, uidess it has been determined very recently, the above method is adajited to finding the time at ship during ihat portion of the day when the single altitude is not ju'aclicable. III. By Equal Altitudes. 790. Since the altitude of a body which does not cliaiige its declination varies exactly at the same rate while rising on the IC. side of the meridian as while falling on the W. side, the same altitude occurs iit the same hour-angle on each side of the meridian, and the middle jioint of time between the instants of two equal altitudes is the instant at which the body passes the meridian. Hence the time and, consequently, the error oi the watch, may be found by observa- tion of equal altitudes. In the case of the sun, the middle ))oint of time, or the mean of the observed times of equal altitudes a.m. and p.m., is apparent noon. In the case of a star, or other C( lestial body, the mean of the observed limes corresjionds to the K.A. of the star when on the meridian, that is, to the sidereal time, which may be converted into A.r. or .MX 288 NAIITTCAT, ASTRONO-tT. 797. Since tlie sun cliangcs liis declination sensibly in iartje intervals of time, two equal alts. a.m. and p.m. do not in gcncrnl correspond to equal liotir-angles, ana it becomes necessary to apply to tlie mean of the observed times a correction, which is called the Equation of Equal Altitudes. The object of the computation is to find what time tlie watch Biievved when the body was on the meridian ; the rate, therefore, does not affect the result, unless it is irregular, in which case the mean of the a.m. and p.m. times is not the time shewn by the watch when the interval is half expired. In like manner, tlie variation of the sun's motion in R.A. (which is the variation of the equation of time) produces no effect, provided it be uniform. The irregularity of this variation is inconsiderable 1. Equal Altitudes at Sea. 798. When the course made good durins: the interval of tbe observation of two equal altitudes is true E. or W., the ship changes her longitude only by the portion of time which she gains or loses on the sun in the interval ; this change introduces no correction, and the only niiestion is tiie time by watch when the interval is half expired. But when the ship changes her latitude, the same altitude no longer corresponds to the same time from noon, and a correction becomes necessary.* 799. Thismethod, though but approximate, has some advantages: it is independent of the terrestrial refraction, provided this remains unchanged in the interval employed ; and the correction for change of lat., when necessary, requires the lat. and alt. to be but roughly known. In the tropics the interval may in general be very small, on account of the rapid change of altitude, and the correction for change of latitude in such cases may sometimes be omitted. In high lati- tudes, on the contrary, the ship's change of latitude considerably alters the time from noon at which the 2d alt. (which should be equal to the 1st) is taken : hence, in such cases, the method is less useful. Note. — As the equation of equal alts, is gener.a'ly a small quantity as compared with the correction due to change of phice, we shall not here consider it. If, however, it is required to introduce it, proceed afterwards to No. 806. 800. The Observation. Observe the sun's alt. before noon, noting the time. Note the instant of the same alt. of the same limb P.M. For greater accuracy, several equal alts, should be obtained. When the motion in alt. is quick, both limbs may be observed. 801. The Computation. (1.) Take the mean of the a.m. and r M. times by watch ; this, when the ship does not change her lat., is the mean time by watch of apparent noon. Then the Equation of Time applied as to Mean Time, will give the time of mean noon at ship as .shown by the watch. Applying to this the error of the watch on Greenwich will give Greenwich time at the mean noon of the ship, which is the longitude in time. ♦ N'.B. — The altitude should not be less than 70°, or the time from noon more than lo". FIXDINCi TIIK TIME. 2S'J (2 ) Correction for change of latitude. With half the interval aa an hour-angle compute the azimuth, No. 676. To the log. sine of half the D. Lat. made good, add the log. .sec. of the lat., and the log. cotan. of the azim. : the sum, rejecting tens, i.s the log. sine of the correction, in time. When the ship has approached the sun in the interval, subtract this time from the above mean ; when she has receded from the sun add it : the result is the time by watch at apparent noon. R.t. 1. June 8th, 1S26, lat. by ace. 6° N., at 2'' 43™ '* hy watch (a.m.) and at 3' °"' 3" (p.m.) libs. alt. Q 84° 30' to the northward ; course, N.N.VV. true, rate, 3J knots. TJie interval, 17"', gives DUt. run I'l mile and D. Lat. i. Alt. (true) 84° 46' sec. i"040 D. Let. 30" sin. 6-163 Decl. 22 50 COS. 9-965 Lat. 6° sec. o'ooi Ualf-Int. 8'"3i' sin. 8-570 Az. 22° cot. 0-394 Azim. 22° sin. 9-575 Corr. — o'' o"' 5" sin. 6-559 a ;i 3^ T. by Watch of App. Noon 2 51 27 or Watch /os<. Here the sun is to the nortliward, and the course is to the northward, or the ship hss tpproavlied the sun. Ex. 2. June 22d, 1828, at sea, lat. 4*^8., course S.W. true, rate 7^ knots, obs. alts of the sun to the northward ; ship receding from the sun. Alt. Q 59° 44 5° 55 is 12 Times 12" 29"' 57' am. 3° 53 3' 4-5 7 6 39* P.M. 37 45 Means 12 30 52 » 7 30 40 int. i" 37'" 5'- The Dist. ran in i" 37" Alt. 60° Dccl. 23i Half Int. 48"- 30- Azim. 221" Approx. T. h m.; D.Lat. 0-301 9-962 9-322 9-585 y Watch of no made good, 8' D. Lat. Lat. Azim. Corr. +0 on I 19 5- 4' 15" 4° 0-4I' 19 16 16 or Watch fast. sin. 7-092 cot'. 0-383 sm. 7-476 T. by Watch of App. Noow j 19 57 or error of tlie watch, fust. 802. Degree of Dependance. (1.) The error of time due to an error of 1' in one of the alts, is half that due to 1' cliange of alt.. No. 788 (I.) (2.) To find the error due to an error of 1' in the D. Lat. made good, divide the correction obtained by the D. Lat. For ex., 1' error •a Ex. 2 causes 5' error in the correction. 2. Equal Altitudes on Shore. 803. The method of equal altitudes is susceptible of consider.ible accuracy, but it can be completely put in practice on shore only, as the sea-horizon is always subject to uncertainty. [1.] The Sun, Morning and Kvenwg. 80.5. The Observation. In the a.m., when the sun is within the limits (No. 778), set the index of the sextant at the altitude, nearly ; chimp the index, and observe the instant of the alts, of both limbs, noting the times. Do the same in the afternoon, when the limbs will follow in reverse order. U 290 NAUTICAL ASTRONOMY. Tlie value of tlie method consists in tlie same altitude being repeated, without regard to the precise measure of it. But as the second or corresponding altitude is often lost by a cloud hiding the object, tilt usual practice is to set the index to certain whole divi- sions, as 10', 20', &e., and to observe the altitudes. The moving of the index destroys, indeed, tiie integrity of the method, since the second altitude is no longer identical with the first, but is merely inferred to be equal to it from the reading. The errors, however are greatly diminished by taking numerous altitudes : or a number of instruments may be employed, set to different altitudes. 806. The Computation. (1.) Reckon the time p.m. as I2\ 13\ &c., instead of O*", 1'', &c. Add together the A.m. and p.m. times of oTiservation ; take the mean of these sums, and divide it by 2. Take the difference between tlie 1st and 3d times (as set down in the txample below) to the nearest minute, and call it the interval. (2.) Find the Greenwich Date for apparent noon at the place; reduce the sun's decl. (p. I. of the Nant. Aim.) to the nearest minute only, marking it as of the same or contrary name to the latitude, and as increasiriff or decreasing. Reduce the equation of time, p. 1. Naut. Aim. (3.) Take the sum of the changes of the suu's declination for the 24'' before and the 24'' after the Gr. Date ; call this the double change.* (4.) Compute the equation of equal altitudes thus: — Part I. From Table 72 take out the logarithms A and B. To log. A add the log. cot. of the latitude and the prop. log. of tiie double change : the sum, rejecting tens, is the prop. log. of Part 1. Part II. To log. B add the log. cot. of the decl. and the prop, log. of the double change: the sum, rejecting tens, is the prop. log. of Part II. (5.) Apply these parts, which form the equation, to the ap])roxI- mate noon by watch, by the following directions. Declination increasing 1 Part I. Part II. Lat. an of the same 1 Declin. of contrary names. luf less than 12 hours. rval greater than 12 hours. sub. add add sub. Declination decreasing add sub. sub. add The result is the time shewn by the watch at the instant the Bim was on the meridian, or apparent noon by the watch, and there- fore shews the error of the watch on A.T. To obtain the error on M.T. To apparent noon, 0'' 0'" 0\ or * As the decl, in Table 60 u given only to the nearest from tins table, may be a minute in error. This will not bf tquiil alts. ; but, for precision, the Nautical Almanac i: minute, the daily change, as taken ause an error of 1* in the ctiuation nectssar;. FINDING THE TIMR. 29-1 12" 0"- «)', apply the reduced Equation of T. as directed p. I. of the Naut. Ahn., or Table 62: the result is the mean time of the sun's iiieridiau passage (as in No. 624). By comparing with this the time of apjiarent noon by the watch, its error on mean time is found. Three places in the logarithms give the equation to O'-l.* iJx. 1. Feb. 15th, 1830, at Ascension, lat. 7" 57' S., long. 14!° W., the following obser va(ioii3 of the sun's limbs were taken in the quicksilver, the sextant being clamped at Xi". A.M. P.M. Sums, deducting 24. id'' +5'°40" 17'' 19"' 19* 4'' 14"' 59" Reil. Decl. 11° 44' S. 10 47 54 17 27 85 4 '5 \'A Eq. ofT. 13'" 50-5 arf<fi<ire. Sum 8 30 1-5 4 15 0-7 Two-Jailychange, 39'i2', .\|.prox. Noon by Watch z 7 3° '3 decreasiny. From .J' 46"' Log. 15 2-412 to 17 27 Decl. cot. 0646 Int. 6 4. log. A 2-218 0-66I Lat. cot. 0-855 Part II. 2-1 3-719 39'iS-i..l. o-66i Int. less than 12'' ; decl. deereming, Part I. 2<o i.i. log. 3-734 tract. l.ai. and decl. tame name ; dccl. decreat- Approx. Noon z' 7'" 30"- 3 my, add. -^-:^ J Eq. ofEq. Alts. -0 -> App. Noon by Watch 2 7 3^^ '^ Eq. of T. additive, or 1 , , ,„ - c Jlean Noon, No. 62-1 } '^ 5° 5 Watch fast on M.T. i 53 39 -7 Kx. 2. July a4th, 1S7S, lat. 55" i' N., long. 0" 6'" W., obtained following observal 6' 5i" I 27-5 I 7" i2'"39'-5... 7 8 57 - Sum g" ' 9- 30-5 19 24-5 ° 55 -o 19 27-5 9 43-7 sextant being clamped 1 ©'s Red. Decl. 19° 51' N. deer. Two daily change 25' it' Eq. ofT. 6"' 14'- 1 addit. Int. io*'2'"; Part I., i5'-6, lat. and decl. same name; decl. decrensiny, add. Part II., I'o int. less than 12'' ; decl. decreasing, subtract ; app. noon by watch, o'' 9'" 58*-3; Eq. of V. additive, or M.T. of Mer. Pass, o'' 6" i4'-i. Watch fast on M.T. qI' 3'" 44- 2. [2.] The Sun, Evening and Morning. 807. Instead of observing a.m. and p.m. on the same day, it i.s oucu convenient to observe on the afternoon of one day and the ruDrniug of the next. yVte Computation. (1.) Take the mean of the times as directed; No. HOG ; this is the approximate time by watch of apparent midnight. Find the interval as in No. 806. (3.) Find the Green. Date in app. time for midnight at the place ' It is often convenient, when all possible accuracy is required, to employ tlie logarithnu imbers. In this case, take the arilh. complements of the logs. A and B, employ tha ciiLs of the lat. and decl., and the log. of the two-daily change in seconds. Fx. (the above.) Leg. A 2-2iS3 ar. CO. 7-7817 IM. tan. 9-1450 39 iS' = i35S' log. 3-372; Part I. !• 99 log. 0-2992 7-5879 9' 3 54' 292 NAUTICAL ASTRONOMY. Jleduee tlie siiii's deel. ami the Eq. of Tiinp, (.3.) Find the double cliange, as before directed. (4.) Compute the equation of equal altitudes, apply the 1st part the contrary way to (5) : the result is the time by watch of a})parent miduight. Ex. Feb. ?2d, 1830, p.m., and Feb. 23d, a.m., lat. 7" 5/' S., long. 14J" W., ob- tained obst-rvatious of equal altitudes. P.M. A.M. Sums(— 12h). 5. ,8.3 . ,,„ , 5 .9 36 10 58 5 10 40-5 10 57 47- 41 36 Watch j-235 0-855 o-fii4 3-704 4'' I 4 > 4 ■ 8 "' 1 9' Deel . 1 2° 44' S. , decreasing. 8 17 Eq. of T. I 3'" 46-4, a<Wi<iBft 8 ,6-5 Apprax. Midnight 65 Part I, 5^ .9"' >8 5 39 log- A- Lat. cot. 43' 8" 4 ' 2 51-5 Double change, 43' 8" 8 .7-5 9 8-7 Part 11. Log B 2-367 Deel. cot. 0-646 0-6.4 2-5 3-627 Tlie int. is greater than ii", that used for log. A being its suppl. The Eq. of eq. alts, is -fo-3 ; the watch fast on M.T. 13.] Equal AlMttdes of a Star. 808. This observation determines the absolute time with much precision and convenience, as there is no equation of equal altitudes. 809. The Computation. (1.) The mean of the times shewn by the watcli is the time by watch corresponding to the sidereal time, or R.A. of the merid., which, in this case, is the same as the R.A. of the star. (2.) Find M.T. by No. 607, and thence the error of the watch. 810. Correction for Change of Refmction. As the method of equal altitudes is capable of much precision, and as the rate deduced may be much affected by small errors in the absolute time, it is worth while to make the proper correction for every cause of inac- curacy. A shift of wind or a fiiU of rain, in the interval, may be accompanied by a change of refraction, which, especially when the altitude is low, may produce a sensible effect. To allow for this, (1.) Find the correction of the refraction at both observations for the barom. and therm., Tables 32,33 ; then, when the corrections differ, (2.) To the prop. log. of their diff. add the prop. log. of the time the sun takes to move through his diameter (which, if not shewn by the observation, may be found by note *, p. 221), and the ar. comp. of the prop. log. of the semi-diameter; the sum is the prop. log. of a portion of time, /ta/f of which is to be applied to the time of noon, or midnight, thus: — 1st obs. A.M., or to the eastward, when the east. refr. is the greater, add; when the lesser, subtract. 1st obs. P.M., or to the westward, when the east. refi. it; the greater, subtract; when the lesser, add. riNDINC. TUK TIMK. 203 El. Mayziat, 1850, Fort VillagnKiioii, R jttained equal alts. 57° in the quicksilver, a.m. :i" If S3 tlian at the west. Reduced deel. zo° 50' N. (of cottt' vy name I El. of T. 3"'44'-7, milr. from A. T ,1c Janeiro, lat. zi" d P.M. ; the rcfr. at 7l.2."'5+' mil ,3^ 25- 6- 13 13 36 2 13 2Z 4 I Correction t ir unequal refraction. 3"' i' '5' 49" Corr.-^:-.] prop. log. 2-95 do. 1-77 Ar. CO. do. 2^ prop. log. 3-65 'Iff), double change 24' 56' The nt. eh from 7'' ii"' tu ■ 3'' »!'" gives the two parts 140 59 I 46 S9'7 I 23 29-8 Appro\. Noon Eq. Equal Alti the equation of eq. ahs. +5«-9. ■Watch 10" i3™i9"-6 ■5 -9 35-7 10 23 34-6 44-7 Corr. for Refract. App. Noon by Watch Eq. ofT. +12" Watch slow on A. T. 1 36 25 4 I Watch slotc on M.T. i 32 40 v 811. Dtgrce of Depcndance. The error of the equation of equal uhitiides caused by an error in tlic double change of decl. is a matter of simple proportion. The etleets of small errors in the lat. and dec!, are insensible, therefore neither the lat. of the place nor the declin. is required to great precision. But variations in the refrac- tion, not to be removed by corrections, will always leave the result iei some degree doubtful. On this account, the method, even under the most favourable circumstances, can rarely be considered as aSbrding extreme precision. IV. Rating the Chronometer. 812. The Rate of a chronometer is the difference of its error from day to day. It is called gaming wlieu the watch goes loo fast, end losing when it goes too slow. 813. When the chronometer is fast, either on G. M.T. or on the time at place, if the error is increasing, the rate is gaining ; if de- creasing, the rate is losing. When the chron. is slow, if the error i.'i increasing, it is losing ; li decreasing, it \s gaming. The amount of the daily rate (supposed uniform) is found by dividing tlie cliange of the error by the number of days in the interval between the observations. Ex. May 27th, at 9'' a.m. chron. slow 2'' 7'" i8« June 3d, at 5*' p.m. slow 2 651 Diff. of Error in 7'' 8i> o o 27 Then 27', divided by 7-33 days, gives 3'7 daily rate, gaining. 814. When the error is found to have changed fiom fast to slow, or from slow to fast, tlie rate is the sum of the errors divided by the n-imlier of days ela])sed. Ex. 1. June 2Sth, at 3 p.m., the chron. was C" 7"o fast ; on July 5th it wa« o» 16" I •low : required the Daily Rate. The sum 23* 1, divided by 7 (days), gives 3*-3, !oti>ig . 20 I NAUTICAL ASTRONOMT;. Ex. 2. On the I4lh, the chron. was o" 17* slow; on the 31SI, it was o" 12' fi&t : required the Rale. The sum o"' 29-, divided by 17, gives I'y, yaimtiy. 815. As the chronometer rarely goes for any lengtli of time ffiihout some irregularity, the rate slioultl lie deduceJ afresh al ,.veiv opportunity. This is done, 1st, by finding tlie absolute error on the time at place, by observaiion, after intervals of a few days ; 2dlj . by direct comparison of the interval of time shewn by the chronometer with tliat measured by a clock of known rate, or with the motion of a star. Also, as longitude is measured by time. No. 479, the absolute longitudes of places, when correctly laid down, and their differences of long, may be employed in a corre- sponding manner. All observations for the purpose of rating a chronometer should be made, if possible, on shore, on account of the uncertainty of the sea-horizon, because a small error in tlie absolute time may produce a great error in the daily rate deduced. Also, the observations should be made by the same person with the same instrument, and under the same circumstances, as nearly as possible. 1. By Comparison with the Absolute Time, or Longitude [1.] Bylhe Time. 816. The best observation (out of the observatory) for the pur- pose, is equal altitudes carried on for several days. The next in value is the same alt. repeated several days successively, in the same part of the day ; for the times determined by a.m. and p.m. sights on the same day do not, it appears, agree exactly either at sea or on sliore.* As the rate rannot be depended upon for a considerable length of time, it is necessary to take frequent opportunities of obtaining alts, on shore by the artificial horizon. It is pi-oper. therefore, to remark, that by a little care, and by not mixing a.m. and p.m. sights, the rate may be determined nearly as well as by oqual altitudes. 817. At sea, the lunar observation, No. 836, or, under very fa- voui'able circumstances, the moon's altitude. No. 864, affords the absolute error of the chronometer on G. M. T., and may discover, accordingly, if any considerable change in the rate has taken place ; but it would be highly injudicious to attempt to establish a rate from obsei nations so discordant as these usually are. 818. An excellent method lias been afforded of late years, of de- termining the error and rate of the chronometer by the establish- ment of time-balls at some observatories. These, with tiie (1. M.T. at the instant the ball is dropped, are given in Table 13. The time- ball obviates the necessity of observations for rate. 819. When the ship leaves any place, and after an interval not much exceeding a fortnight returns to it again, the error of the * The late Captain Hewett mrormetl me, that being obliged to keep account of the daily ra'i'R of liis chronometers, by means of altitudes observed from the sca-horizoii, while sni . reying the North Sea, in 11. M.S. Fairy, the constant discrepancies betwceu the ( u. and r M sighls rendered it necessary to employ the A..M. sights aluiie, TIN 1)1 NT. TIIK TI.MF. 295 chninonicter acciiiiiulated in licr absence is found directly by com- .liirniy: the time shewn liy the clirononieter with tiic times olilaincd by ohservation both at lier ilci)artiire and at her return. The error thus found affords the actual sea-rute, and tlie method, when it <-iiM be practised, is far more etticieiit than tiiat of deducing iiarhour rati's. Ex. By an observation taken immediately liefore the sliip'.s departure from a port tlie cJiron. was found slow 3'' 27'" 14". By an obser^-ation taken at her return, or 11*3 <1hvs afterwards, tlie error was -^^ 27"' 44*-5, or ^o'-j more. Hence the Rate during her absence lua been, on the average, 2"7 losiiiy. [2.] By the Lonijllnite. 820. Wlien, on making a well-determined point of land, the long. by ('hron. does not agree with the actual position of the ship, and when, accordingly, the chronometer must have been going at a ditt'erent rate from what was supposeil, it w ill be convenient to refer to the following Table. The land made unexpecteiily. Sailing W. The Chronometer has gained less, lost more. gained more, gained less, or or lost less, lost more, than allowed for. gained more, or lost less, Kx. A ship from India to the Cape of Good Hope makes the land unexpectedly. The ship is sailing \V., the land made too soon ; the chron. has therefore gained less or losi more than allowed for. But it must be borne in mind that chronometers do not preserve the same rates, generally speaking, for a long time together; and, therefore, after a considerable interval, as upwards of a fortnight, this method shews only the gain or loss uii the whole, not whether the. chronometers are gaining or losing now. 2. By Comparison of Intervals, of Time, or Lomjitude. [1.] By a Clock. R'21. The clirononieter being compared at different times with a clock of which the rate is known (as in No. 504), the difference of the errors for the intervals is obtained, and tlience the rate is deduced. The mode of comparison is already described, p. 203. [2.] By a Star. 822. Since every star returns to tiie same point of the heavens 3" o.VOl of mean time earlier every mean solar day, the ret'irn of the same star to the same altitude, or to the wire of a fixed telescope, (lay after day, determines the rate very cjrrectly. The alt. should 29G NAUTICAL ASTRONOMY. he considerable, in order to avoid errors of refraction, and the Icleseope, for the same reason, should be nearly in the meridian. To find the rate, multiply 3"" 55'-91 by the number of days elapsed, and subtract the product from the first time noted ; the re- mainder is the time the chronometer would shew if it went uui- forndy, and the difference between this and the time it shews is tiie difference of the error for the interval, which gives the daily rate. Ex. At an observation of a star on May ist, tlie chron. shewed 7'' 51™ i !• ; after foul days it shewed y"" 35"' 44"'6 : required the Daily Rate. First time noted 7'' 5 1"' 1 1' 7 35 =7-4 Gaining in four days 17-2 hence the daily rate is 4«- 3, jainiBy. The disappearance of a star behind any elevated object answers the same purpose. [3.] By Difference of Lnng-tude. 823. When the error of the chronometer upon the time at an^ known place A is compared with the error on the time at another known place B, the difference between these two errors is the ditt". long., in time, between the places. Hence if the difference of the errors does not agree with the Diff. Long, found from Table 10, or in Table of Secondary Meridians, p. 392, the discrepancy arises front a wrong rate having been employed in the interval between the observations for time, and the true rate may be found by trial, aa in the following example: — Ex. At Falmouth, Feb. 3d, at 3'' 20"" i8» M.T. by observation, the chron. shewed 4- 3 1'" 47', or was I '' 1 1"' 29" fast. At Funchal, on the 12th, at si" 30"' 27' M.T., or 9' i days afterwards, the chron. shewed 7'' 29'" 34*. The supposed rate, 2'-3 gaining. Tlie D. Long, in Table 10 A is 47" 28*. Required the true rate. Obs. at Falm., T. by chron. 4i'3i"'47' | Obs. at Funchal, T. by chron. 7'>29"'34« M.T. by obs. 3 20 18 2' 3x9- id. yam -21 r, fast I 1 1 29 I 7 M.T. by obs. 5 fast I 58 46 , ditto I II IQ Difference, or chron. I). Long. 47 17 This diff. should be 47" 28", or is too small by 1 1». By inspecting the process, it is ■fijent that the quantity 2i« (which, from the nature of the case, is supposed to be in error) is too large by 1 1". The rate, therefore, is lo' divided by 9-1, or I'-i gaining. When one error is fast and the other slow, make tlicm both fiist or both slow, by adding or subtracting any number of hours. 3. Keqnii(/ Account of the Chronometer. 824. In keeping account of the chronometer, the error on G.M.T. is entered in a book as fast or slow, with the date, and tiie rate is applied to this according as it is gaining or losing, day by dav. If, after a timr, the long, or (i.M.T. lie obtained independently, liii error ou G.M.T. is foiuui ; if this doe.s not agree will' tlic rate FlXinXO THE I^OXGITI'DK. 207 allowed, a new rate must be assigned from consideration of tlie «!ircumstances. 825. As it is impossible, without an independent reference, to determine whether a ehi-onometer, A, is gaining upon another, B, or B is losing while A goes as before, no direct rules of certain application can be given for reducing the rates of chronometers by mere comparison. Since, however, it may be presumed, in general, that in a number of watches the true time will be that shewn by the majority, regard being had to the quality of each, it is proper to keep au account, in which an approved watch being taken as the standard, the rest are severally compared with it evi>i-y day. It is convenient to distinguish the chronometers by letters, as A, B, C, &c., and to write the difference between A and B thus, A — B; that between A and C thus, A — C, over each column. Advantage should be taken of favourable opportunities of land- ing at well-determined places (sec Table of Longitudes accepted for Secondary Meridians, p. 392) for good observations of time, bi'cause the diff. long, between the places will at once discover any considerable change in the rate, afford means of correcting it, and be a means of obtaining the sca-ratcs of the chronometers. CHAPTER VII. Finding the Longitude. I. Bv THE Chronometer. II. By the Lunar Orservation. III. By THE Altitude of the Moon. IV. By an Occultatjon. V. By Eclipses of Jupiter's Satellites. 82G. The apparent motions of the celestial bodies parallel to the equator, produced by the revolution of the earth round its axis, being perpetual, no fixed point or circle can be obtained from which the longitude of the observer, which is measured, like right ascen- sion, on the equator, may be determined. Longitude, accordingly, can be ascertained only with reference to the meridian of some other place ; and, as it is measured by time (No. 103), it is determined by comparison of the time at place with the time at some other place. I. By the Chronometer. 1. Determination of the Absolute Loncjitxide. 827. The most convenient method of finding the longitude is by comparison of the time at place with the time at Greenwich, as shown by a chronometer. The mean time at place being found (Chapter VI.), take the difference between this time and tin- time by chronometer, brought up to the time of observation by applying the error with the rate. i;l)8 NArTfCAIi ASTRONOMY. WIjcti tiie time at Greenwich is the least, the lonp^. is 13.; \\\\fn the greatest, it is W. Rt. 1. TheM.T. at place is 5>'4S"'2"; the G.M.T. i§ 4" 15™ ii': hence the Long. o» the place is o'' 27'" 9", or 6° 47' 15" W. Ex. 2. The M.T. at place is 7'' I4"'22«; the G.M.T. is 2'' 6"' 57': hence the Long, is 5" 7"'2i;», or 76"' 51' 15" E. 8'28. Degree of Dependnnee. Tlie time at place, as deduced from ohservation, and tlie time shewn ))y chi-on., being both liable to error, tiie error of tiie resulting- longitude is made up of the sum or ditfei-ence of tiiese two errors. 829. When the r;ite of the chronometer has changed, and the long, is reqniied at a time past, the error of the chronometer at the lime proposed must be deduced from the two rates by consideration of the circumstances, as no rule can apply to all cases. 2. Determination of Difference of Longitude. 830. The ordinary nietiiod is to find the absolute longitudes ot both places by coiiinarisoii of the Greenwich mean time, as al)ove desflribed, and then to take tiie diti'erence between them. Ex. M.T., at a place A, is 3" 11"' 45', when the G.MT. is 7'' 7"' 18": hence the Innj i.f A is 3'' <;V" 3S' W. Again, some days afterwards the M.T., at a place B, is 2'' 19'" 45". wlien the G.M.T. is 6'' 26'" 34": hence the long, of B is 4i'6'"49«W. Tlie UiFK. Long, between the places is, therefore, 11'" 14", and B is west of A. 831. But it is more concise, in a question relating to a difference. only, to proceed without regard to the absolute longitude of either jilace, by considering merely the error of the chron. on the time at each of the two places, as in the following example : — Ex. 1. At 3'' I !■" 43' M.T., by obs. at a place A, the chronometer shewed s"" 1 1"' 19', or was 1'' 59"' 36' fast on the time at A. Again, some days afterwards, at 2'' I9"'45' M.T., iX a place B, the chron. (after applying the rate) shewed 4'' 30'" 35', or was 7.^ lo'" 50" fast on the Now it is evident that if A and B were in the same long., the chron., supposing the rate truly determined, would have the same error at each place ; and hence the dilference of the errors, i'' 59"' !6» and 2'' 10'" 50', or 11"' 14', is the Diff. Long. Since the chron. is faster at B than at A, the time at B is be/iind that at A, or B is west of A. Tlie ]iroceeding, reduced to a rule, is as follows : — Find, by observation, tlie error of the chron. on the time at jdace. Having moved to anotiier place, take an observation for tini<!; correct the time shewn by the chron. by applying the rate for tiie time elapsed since the former observation, and lind the error; til'! difterence of the two errors is tlie diti'. long. When tiie chron. is fast at both places, the place at whicii the error is the greatest is west of the other. A\'!ien the chron. is slow at both places, the place at wlfudi the error is the greatest is east of tiie other. When the chron. is fast at one place and slow at the ether (as uiaj occur >'-heu the error is less tliaii tin: di"'. long.), add 5 or ti Fi\niN(i THE i.oNCiiTrnR. SDD •s t(i onch of fill' times l)y cliron. in order to render bolli tli« rs of the same kind. Ex. 2. At A, M.T. s*' 36" 'o't chroii. 6>> 36" 20«, error i"" o"> lo" fast At B, M.T. 3 28 30, chron. 4 9 20, error o 40 50 A west of B, DiFF. Long, o 19 20 Ki'l. Siiipo tlie wliolc value of a elironometric deteniiiiiation d('|i('iids ii])oii the rate of the chroiiouu'ter, and since the rale is lia'jle to change, the result is better as the time occupied in the run IS less. This, however, does not, in strictness, apjily to inteivak less than 24 hours; for the works go through an entire revolution in 24 hours, and the rate, which is determined for an entire day, may be unequally distributed over different parts of the 24 hours. Tor extreme precision, tiie rate should be known for given intervals on the dial-])Iate. H'-i'-i. Wlien the ship returns vvitliout loss of time from a place to that from which she set out, the opportunity will in general be veiy favourable for determining the ditference of longitude. 834. While a chronometer continues to gain or to lose, the ditfer- ence of longitude shewn by it between two places will be dirt'cicntly affected, according as it is measured eastwards or westwards: hence, if the differences do not agree, the true diff. long, will be between them. When the chron. gains on its rate, flic computed long, is to the west of the true long. ; when the chron. loses on its rate, the coni- jiiited long, is to the cast. If the rate is steady, the true ditf. long, will be coriccfly found by dividing the error according to the number of the days in tli»~ two j)assages. 3. Cuinmunication of Chronomclric Differences. 8-3.5. Individuals possessing one or more good chronometers fre- quently have op])ortunities of furnishing, verifying, or correcting meridian distances. It is proper, therefore, here to enumerate the considerations wiiich ir.Huence the value of the results, more espe- cially as many such determinations arc communicated to authority from time to time, which, liowever, not lieing accompanied with the details necessary ibr an estimation of their value, leniain uueui- ]d()ye(i. (1.) It is absolutely necessary to specify or to (.'escribe tlie cxacl spot of observation at each place. (2) The number of dai/s employed in the run, or in the interval between tlie observaiions for time, or both, if these differ much, togetiier with the number of chronometers, should lie expressed ; also, I'le imies and manner of rating, and the character of the rate, as steady or unsteady, should be bricHy noticed. (3.) Tile maker's name and the number of the chronometer should be specified, because tiic character of a watch alliuct8 the value «f * •lctcruiiij;ition in which il is eiiijiloyed. 800 NAUTICAL ASTRONOMY. (4.) When tlieic are several chronometers, the result given hj each should be exhibited. The general arithmetical mean should be given, and, besides this, an estimated mean, obtained by giving more or less weight to the several results, according to the performance r,( each chronometer, and of which the observer alone can be a judge. The two final results should be expressed in ti.me, and also in arc, for the more ready comparison of positions on the chart. (5.) The e.rlrriur dijfrrence of the greatest and least results by the different cliiDiioiiiiters employed should be stated, as this shews wliether the ciuononieters went well together or not; for, though their going together does not prove that all or any of them are right, their not going together proves that some of them are wrong. (6.) All observations for the longitudes of places are supposed to be made by nieiins of tlie quicksilver, unless the contrary is ex- pressed. Wlien the altitudes are taken from the sea-horizon, tlie result should, tlierefore, be distinguished by the word (sea). (7.) It will be useful to state the temperature of the chronometer- room, and to remark whether it has remained constant or been subject to variation. Also, the general direction of the ship's head should be noted. (8.) Lastly, every result should be given without any regard as to whether it agrees or not ivith received determinations. Many received positions are very erroneous, and the only means by which they can be decisively rectified are the comparisons of independent and impartial evidence. In the following example, D. L. is the abbreviation of DiflT. Long.; ch. is that of chronometers; d. that of days; and the ex- treme difference is denoted by the number of seconds enclosed in brackets, implying limit or boundary.* Ex. May, 1838, Capt. A., of H.M.S. , sailed from Barbadoes to Port Royal, Jamaica, the points of observation being Engineers' Wharf and Fort Charles. He carritd live chronometers, viz., No. 152, Molyneux ; No. 191, Breguet ; No. 702, Arnold and Dent; No. 650, Parkinson and Frodsham ; and No. 490, M'Cabe. The passage occupied seven days. The extreme difference of the results was 7 seconds of time. The arithmetical mean was i''8'»49»; the estimated mean, i''8"'52'. The temperature of the chronometer, room ranged from 78"^ to So" ; tiie ship's head chiefly west. These particulars, abbreviated, stand thus : — Oapt. A., May 1838, D. L. Barbadoi, (Eng. Wharf) to Port Royal (Fort Charles), 5 ch. 7 J- [7'] Aritli. Mean, ii' S"'49"= 17° 12' '5 Estim. Mean, 1 8 51 =17 13 o p. 402, to which the reodei u M. B. A. and D. P. and F. M'C. No No No. No No. 152 i^S 702 I 8 650 I S 490 I 8 •46' S3 Temp. 45 Head« 49 ni" to est. * This plan referred for otb was proposed in the Naut. ;r details of the subject. Mag. KINDINO THE LONOITUDR. 301 II. Tm: Lunar Obskkvation. Cl««nni; the DisLince, Nos. B42, 344,845 — Lunar Obs. by the Sun. No. 847 — Lunar Obg by a Star or a Planet, No. 849— Special Corrections, No. 851— Degree of D-jpemUnce, No. 858 — CiJeulation of Altitudes, No. 803. 836. The aiioiilar distance of tlie moon from any celestial body liciiiL^ in per])etual change, each of the several degrees of magnitude throiigli which it passes corresponds to a certain instant of time. .Accordingly, the distance of the moon from the sun and certain other bodies, at the end of every three hours, being given in the Nautical Almanac, the observation of this distance atlords the means of determining the time at Greenwich, and thence the longitude of the observer. This observation, on account of its great importance at sea, has been distinguished by Ihe name of the Lunar Observation. 837. If the distance between the moon and the other body were the same to the spectator, whether he were at the surface or the centre of the earth, tliere would evidently be nothing more to do tlian to measure the distance by an instrument, to find from the Nautical Almanac the Greenwich time corresponding, and to com- pare this time with the time at place. But the refraction of the sun, a star, or a planet, being greater than its parallax in altitude, causes one of these bodies to appear above its true place ; while, on the contrary, the moon's parallax in alt. being greater than her refraction, causes her to appear beloio her true place. Z is the zenith, S and B the true places of the sun (or star) and moon, S' and D ' their apparent i)laces. Then S D is the tnie distance, and S'l>' the aiiparent distance SS' is the sun's corr. of alt., dd' the moon's corr. of alt. In fig. I, where the j's alt. is the lesser, the app. (list, exceeds the true, for d' is fartiier from S than 5 is, and S' is also f;xrthcr from D than S is. In fig. 2, the app. dist. is the lesser. In fig. 3, both angles at S and D are acute, as is tlie case when the alts, are nearly erpial, and aiuays wlien tiie distance exceeds 8o°. 302 NAUTICAL ASTRONOMY. As 5 5' is always less than 5G', the arc » m, fig. 3, of a circle, having its centre at S, is nearly a right line, and j'w (which, from tiie apparent place of the moon, is here the excess of the app. dist. aho^e the true) is equal to ]>]>' cos. of tiie angle at ])'. The like rerm (or 1st correction of the app. dist.) for the sun is SS' cos. S, or SS' COS. S' nearly. This is the principle of the approximate nietiiods.* Hence the apparent distance between the moon and the other body differs from the trxte distance, e.Kcept in the particular case in wliich the two opposite effects happen exactly to compensate. Tiiis last circumstance may sometimes occur during the time that two bodies within distance are above tlie liorizon, but not being discover- able from the observation it is productive of no simplification. Tlie process of reducing the apparent to the true distance, or removing the effects of parallax and refraction, is called Clearing the Distance. 838. It is evident from the above that the difference between the true and apparent distances depends almost entirely on the cor- rections of altitude (No. 438) ; and, consequently, is affected by every variation, however minute, of those corrections. Also, since the most rapid change of distance is about 1* 48' in three hours, tlio effect of 1' error of dist. is 25' of long., or the effect of 15" error of distance is 6' of long., in the most favourable case. Hence it may become of great importance to the accuracy of the result, in many cases, that the heights of the barometer and thermometer should be noted at the time of observation. 839. Tlie lunar observation, wliich is tlie only independent me- thod of finding tlie longitude generally available at sea, is also, from not being confined like some others to a particular instant, of service on shore. A single observation, however, is not capable of affording a decisive result ; great practice is necessary for measuring the distance successfully ; and the application of so many small cor- rections as are necessary wlieii accuracy is required is, even with extraordinary care and some skill, scarcely compatible with extreme precision. 840. Limits. The distance must fall between the greatest and * The approximate process will *>•■ easily intelligible by attending to the following considerations. The moon must always be raised, and the sun or star lowered, to attain their true places. Now, when the moon is the lower of the two bodies, it is evident that raising her will diminish the apparent distance ; that is, her correction of distance must be sublractive. Again, when she is the higher body it is generally additive. When the sun or star is the lower body, lowering it will increase the app. dist. ; its corr. of oist. is therefore additive, but subtractive in general when the uppermost body. The angle at the lower body, Z j'S', orZS' j', is always acute, the corresponding angle at the other body will generally be obtuse when the altitudes are very unequal, and the dist. not great. The correction of di»t. in Method 1. is the D. Lat. corresponding to this angle as a roiirse, and the corr. of alt. as Dist. The sum or dift'. of the Dep. and N is the cosine o( the angle in question to the radius 100. When the dist. is less than 90' and the Dep. greater than N, the angle is acute, but obtuse when the Uep. is the lesser. Thus, in Ex. 1 the angle at the moon is 55°J ; that at the star, 76°. When the moon's alt. amounts to nearly 80", or when the distance is so small as 20 , M uid N vary irrcgu irly, and Method I. does not serve wcU. PINniNG TIIR I.ONCilTtTDF.. 303 If-ast (listaili'cs in tlic A"aii(icai Almanac. The alls, slionld not Lo less tlian 5° or iP : ami, when tire liaromctcr and tliernioniet<^r arc iioi at liaml, nut less than 12° or 15°, t'spucially hi very hot or very eul(i woathcr. As the chief jiart of tlie eompntation consists of clearing the distance, it will bo more convenient for reference to consider thin portion of the work separately. 1 . Clearincj the Distance. [ 1 .] Approximate Metliods. 841. In these methods the object is to find the correction of tlie apparent distance due to the corrections of altitude of each body. The first, or that by inspection, is performed by means of the Spherical Traverse Table ; and the second, by logarithms* is a useful and convenient process, without the embarrassment of various cii-ses, and renuiring oidy four ])laces of figures. The a|)pro.\imate methods are, in general, not susceptible of much ))recision when the distance is less than 2(>°. 842. Method I. By Inspection. (1.) For the Moon's Correction of Distance. With the moon's app. alt. and tlie conipl. of the app. dist. to 9()<», take out M and N. With the sun's or star's-f- alt. as Course, and M as Dist. find the Dep., which place under N. When tlie distance is les<t than 90°, fake the difference of this Dep. and N, marking the Dep. according as it is greater or less than N. When the distance \s greater than 90", take the sum of the Dep and N. With the Dist. 100, and the said diff. or svm as D. Laf., find the Course. J With this course and the moon's corr. of alt. as Dist., find the D. Lat; this is the moon's correction of distance. For the Moon's 2d Corr. Enter Table 56 with the app. dist. and the moon's corr. of alt., and take out the seconds. Enter again with the corr. of dist. and take out the seconds. The diff". of these two quantities is the 2d corr., which apply as directed in the Table. (2.) For the Sun's or Star's Correction of Distance. With the sun's or star's app. alt. and the co-di.st., take out M and N. With the moon's alt. as Course and M as Dist. find the Dep., which place under N, marking it as greater or less than N when the .list, is less than 90°. Take the diff. or sum as before directed. With the Dist. 100 and this diff. or sum as D. Lat. find the Course. With this course and the sun's or star's corr. of alt. as Dist. find the D. Lat.; this is the corr. of distance required.?) * lliia is a slight variation of tlie melhod commonly known among leurtD as Nurii't ItC iiillioU, ana attributed to Mendoza Rios. t In the case of a planet, substitute the word planet for star in the several rules. X if this sum or ditf. exceed 100, a mistake has been made. t I'he correction of distance may be found more correctly by multiplying the ililf. or wui 30t NAUTICAL ASTRONOMT. Note. In finding tlie moon's oorr. work to the nearest half degree ; and when the SLii's or star's alt. is less than io°, take out the Dep. to the nearest third or quarter of a degree. In the sun's or star's corr. work to the nearest whole degree. Apply the corrections to the app. dist. as follows; the result is the true distance. Distance less than 90° Distance greater than yo° J Corr. of Dist. 01* Corr. of Dist. J Corr. of Dist. or* Corr. of Dist. When th less thanN e Dep. is greater thanN When t less thanN e Dep. is greater thanN sitb. add add sub. sub. add ■5, 2d Correct, of Dist. add 5 2aCoriect. ofDUt. »«4. Ex. 1. (Dist. less than 90°.) App. alt. 47° 3"'; A. alt. J 36° 52'; app. dist. 4.8° 20' 29". © corr. of alt. 47" ; J corr. of alt. 45' 35". (Co-dist. 414°.) J 's Corrections. J 37° and 414°, M 167, N 66-5 47^° and 167, Dep. 113-1 (gr.) (</#■) 56-5 Dist. 100 and D.Lat. 56-5 give the Course 55^ ; at which, Dist. 45' gives D. Lat. 2$'$ 25' 3°" 35" ^ 25 50 (which sub., since 123-1 exceeds 66-6.) J 2d corr. i5" 5 ■^ •>" E.x. 2. (Dist. greater than 90°.) App. alt. 13° 10'; app. alt. J 36" 6'; app. dist. 120" 29' 53". Hor. par. 59' 42" ; J corr. of alt. 46' 5S" ; © corr. of alt. 3' 56". (Co-dist. © 13° and 30°, M 118-5 N 13-3 J 36" and 118, Dep. 69J. («*m) S2-7 100 and D. Lat. 82-7, Course 34". 3' gives 2'- 5, 0° 2' 30" 56" 46 ®'s Correction. © 47° and 41°, M 194-3, N 93-2 J 37° and 194, Dep. 1168 (gr.) (.diff.) 23-6 Dist. 100 and D.Lat. iy6 give the Course 76° ; at which, Dist. 47" gives D. Lat. 1 1" (which add, since ii6-8 exceeds 93-2.) © corr. +0° o' 1 1" J 2 corrs. —25 39 -o 25 28 A. dist. 4^ 20 29 True Dist. 47 55 ' J's Con J 36'' and 30.^'', M 143-4, N 42-8 © 13!° and J43-4, Dep. _32^ (sum) 75-7 100 and D. Lat. 75-7 give Course 41'^. 46' gives 34' -7 34' 42" 58" 44 -35 26 J 2d corr. 120° and 47', 1 1" I „ 35, 6 ) -5 + 3 16 —0 3'; 31 —0 32 15 120 29 53 True Dist. '9 57 3S (of the Dep. and N) by the correction of alt., pointing off two more decimals than the product contains. The seconds may either be taken separately, or as decimals of a minute. This process, worked however rouglily, affords a check against a mistake in using the Traverse Table. Ex.1 of No. 842. Di«". 56-5 J corr. of alt. 45' 35 " 45-6 Prod. 2 576-4.J Pointing off two dec. 25-76 Diff. 23-6 © corr. of alt. 47" 47 __ Prod. 1109-2 Poiiitiug olf two dec. 11-09 •>' "' FINDING THE LONCITUDR. 305 Ex. S. App. a.t. 3 72°o ; npp. alt. J 17° ;'; app. Uist. 71° 18' 32", .-orr. of alt. )i": J corr. of alt. 46' 30". (Co-dist. I7i°.) J 17° and 174", M 117-4 N 15-6 ®72''»ndii7, Pep. 111-3 9S-7 100 uBd 95-7, Course 17". 46' 44 o" 30" rg -44 29 I 2d eorr. 72° and 46', 7" 1 „ 072" and 17°, M 3;,8-4, N 94-1 J 17" and 33X, Pep. 153-0 5ii-9 100 and jX-g, Course 54°. 16 ', © corr. + o" o' 9' -o 44 2il — c 44 19 72 18 jr TbCE DlST. 71 34 13 Ex. 4. (Correcting for the barom. and therm.) Suppose, in Ex. 2 above, the barora. i» 3C-7 in. and the therm. 38°; the J corr. of alt. will be 46' 54", by No. C55, and the a'* KcfT. of alt. 4' 9", No. 051. 46' 34 4^' 4' 3' •»" 54" 4J 9" 7_ -3; ^3 +3 ^5 True Pist. 119" 57' 50'' 843. Tlie following examples cxliibit those steps only wlilcli, in proceeding by No. 842, a practised coni]mter will find it necessary to write ilown. The errors are marked against each result as given in Dr. Inman's " Navigation." Ei. 1. © A. alt. 25° 20'; J A. alt. 25'' 35'. ' 21" i app. dist. 104" 37' 49". (Co-dist. H-i^.) of alt. i' 52" ; J corr. of alt. (5»°) 29' 36' [} -29 49 -o 28 36 '04 37 4 9 True Pist. 104 9 13 (3'toosinkUX Ex. 2. A. alt. Spic. Virg. 48° o'; A. alt. J 69° 48'. ^ corr. alt. si''; J ditto, 18' 39", H«r. par. 55'; A. dist. 55" 46' 34"- (Co-dist. 34".) 35^-7 .85-3 262-3 ,80.3 ii:--? >3' 54" 77-0 94- » 30 +0° 0' 48" — 14 24 f\ +2" —0 14 22 -0 13 34 55 46 34 True Dist. 55 33 ( 19" too small) El. 3. A. lilt. © 60' 39 ; A. alt. J 34"4''- ® O". alt. 28"; J corr. 43-40". Hor Vai. 54' 47"; A. dist 43 44 50". ^ Co-dist. 46-.) 175-1 711 296-9 186-8 .70-4 35' i» 82-0 16-4 11 -0° 0' 4" - 35 45 rv-" -0 35 33 -0 35 37 43 44 5° True Pist. 43 9 13 17" too smaUt It iK evident from these examples, whi'-h.wiih those before given, 306 NAtJTICAT, ASTRONOMY. exhibit a surticient variety of cases, that the method is accurate enough for navigation in the open sea. 844. Method II. By Logarithms. — Set down in order tliC sun's or star's ap|j. ah., the moon's app. alt., and the app. dist. ; take half the sum, and subtract from it the first term in order (sun's or star's alt.); call the rem. the 1st rem.; subtract the second term in order (the moon's alt.), and call this rem. the 2d rem. For the 1st Corr. To the log. cos. of the moon's app. alt. add the log. sine of the app. dist., the const. 9-6990, the log. sec. of the half sum, the log. cosec. of tlie 1st rem., and the prop. loj;. of the moon's corr. of alt. ; the sum (rejecting tens) is the prop. log. ol tlie 1st correction. For the 2d Corr. Take the difference between the moon's corr of alt. and the 1st corr. Enter Table 56 with the app. dist. and the moon's corr. of alt., and take out the seconds. Enter again with the above difference, take out the corresponding seconds, .-rnd subtract them from those taken out before : the rem. is the 2d corr. Apply this corr. as directed in the table. For the .3d Corr. To the log. cos. of the sun's (or star's) app. alt. add the log. sine of the app. dist., the const. 96990, the log. sec. of the half sum, the log. cosec. of the 2d rem., and the prop. log. of tiie sun's (or star's) corr. of alt.: the sum (rejecting tens) is the prop, log. of the 3d correction. (As the 2d, 3d, and 4th logs, are common to the two corrections, it will be convenient to take the sum of these three logs.) Subtract from the app. dist. the moon's corr. of alt. and the 3d corr.; add the 1st corr., the sun's (or star's) corr. of alt., and apply the moon's 2d corr. as directed in Table 56: the result is the true distance. Ex. 1. App. alt. 4-7° 31 ; app. alt. J 36° 52 ; app. dist. 48° 20' 29". Sun's corr. of alt. 47"; moon's corr. of alt. 45' 35". ©Alt. 47° 31' COS. 98296 J Alt. 36 52 COS. 9'903i Dist. 48 20 sin. 9'8733\ A. Dist. 48° 20' 20" 5 Corr. Alt. — 45 35 3d Corr. - 37 47 34 ■7 1st Co. + 19 4'! & Cor . Alt 47 J ■M Corr. + True Dist. 47 54 59 '3^ 43 9'699o; 9-9690 HalfS. 66 21 sec. 0-5967' 1st Rem. 18 50 cosec. 0-4910 2d Rem. 29 29 cosec. 0-3079 45' 35"P'--Iog-fJ9^5 47" pr- log- 2-3613 1st Corr. 19 45 pr. log. 0-9596 3d Corr. 37", 2-4678 Diff.* 26 A. Dist. 48°, and 3 Corr. Alt. 46', Tab. 56, 16" 26 6 2d Corr. + 10 i Ea. 2. App. alt. O 32" 36', app. alt. ? 65° 22', app. dist. 81° 15' 51"; Q's eorr. of rU. i' 22", I) 'e corr. of alt. 22' 27" ; n-quirod 'I'nie Distance. 1st Corr. 37", 2nd Corr. o, 3d Corr. o. True Dist. So° 55' 23". Ex. .3. App. alt. » 50° 44', app. alt. J 27° 50', app. dist. 93° 9' 6", ]) 's corr. of alt. 50' 25", * corr. of alt. 47". 1st. Corr. 4' 45", 2d Corr. o, 3d Corr. 9", True Dist. 92° 24' 4". * Tliis difF. is the moon's corr. of dist. by tlic method No. 842. The sun's or s'ar's corr. ul dist. is I'oimd in like manner, tlius ; 47"-37"= 10" (agreeing within l"). riNDlNfi THE r.ONOITUUK. .-^n: [2.] T/ie Htgoroiis Method. 84.1. In this iiietliod wo fiiul, bv <'Hlciilation, tlic true distaiu;e dirt-ctly from tlie apparent distance and apparent altitudes. (1.) Take both the app. aits, to the nearest even or odd minute, take tlieir sum, and call tlie supplement of it the Ist sujiplemmt . Subtract from this suppl. the moon's corr. of alt., and add to it the sun's or star's corr. of alt. ; call the result the 2<l supplement. (2.) Take out the Logarithmic Difference, Table 73. Take the app. dist. to the nearest even minute. Mark the seconds, if taken in excess, to be subtracted, or if omitted, to be added afterwards. To this add the 1st suppl., take the half sum, and from llie half sum subtract the aj)p. dist. Add the loij. sines of ihis half sum and remainder to tlie log. difF. ; flic sum (rejecting- tens) is the log. sine square of an auxiliary arcar. (3.) Under x put the 2d suppl., take the sum and the diff., and half the sum and half the diff. Add togetl-.er the log. sines of the last two terms; the sum is tlie log. sine square of an arc, which becomes the true distance ou ajiplying the reserved seconds. E.^. 1. A. alt. 47° 31'; «PP- alt- 3> 3« ' 52 ; app. dist. 48° 20 29 . Sun's corr. of alt. 47"; moon'g H. P. 58' 3 5"; mcm-scorr.ofalt 45' J5 © Alt 47° 52' > io. 36 52 84 2+ 1st Sup. 95° 36' 0" ■ 45' 35"+ 47". -44 48 X 7 5° 48' 48" 2d Sup. 94 5' '^ Sum 170 40 Diff. 19 2 24 2d Sup. 94 51 12 HalfS. 85 20 HalfD. 9 3' >^ sine g-998';5g > 36" :;o', H. P. 58', 9-99579=' 2. ~ S\ siue 9-218363 pt^. for" 149 35". -45 ©47°. -!4) -64 Log. Diff 8'9957i8 47 54 3^ add 29 TB.Dm. 47 55 1 sm. s<i. 9-217070 A. Dist. 48^20' (19" omitted) 1st Sup. 95 36 Sum 143 56 HalfS. 71 58 sine 9-978124 R.-m. 23 38 siiic 9-603C17 X 75° 48' 48" sin.sq. 9-576869 846. It is useful to bear in mind, as a cliock -against a gross mistake in clearing the dist., that the true and apparent distance- cannot differ by more than the svm of t'ne corrections of altitude. A<r:tin, when the moon's alt. is erjur/t to, or less than, that of the dist. tir other body, the true distance is less than the ap contrary does not always hold when the moon's a 2. Lunar Ohserrathm hy the San. 847. The Observation. (I.) The alts, of the sun and moon an required at the instant at which tlie distance is observed; wIihii llicrcfore, the observer has assistants provided with pioper watche? Ilii-y will obtain the alts, during the time that he is obuerving th rtiblance. See N<js. .OfiU arid r>GI. SOS NAUTICAL ASTRONOAIV. Wlion the observer is alone, lie will first observe the alt. of Ihe body farthest fro-n the lueritlian, then that of the other body, and then the distance ; concluding with the alts, in the reverse order • As precision is not necessary in the alts., one observation of the alt. will generally be enough at each time. Tlie time by watch is, of course, to be noted at each contact. (2.) To observe the distance. Set the index nearly to the dis- tance in the Nautical Almanac, at the nearest estimated Greenwich time; put down one or more shades to screen the central mirror, direct the sight to the moon, and, holding the plane of the instru- ment in the line joining the two bodies, vibrate it slowly round the line of sight as an axis till the sun's image is seen. Make a contact roughly, clamp the index, put in the telescope (previously adjusted to distinct vision by the moon), and complete the contact. See :)ote <;,, p. 182. The relative brightness of the object and image is most conve- niently adjusted by altering the distance of the telescope from the plane of the sextant by means of the screw for the purpose, as this motion causes a greater or lesser cpiantity of light to proceed to the eye from the silvered or brightest part of tiie mirror. Observe at least 3 or 5 distances, or, with the circle, 3 or 5 pairs. When, at sea, the ship has much motion, the observer fixes him- self firmly in a corner, or lies on his back on the deck, in order to remove, as much as possible, the sense of bodily effort and inconve- nience which disturbs the eye and the attention. (3.) For precision observe the moon's true bearing; if she is near the zenith, observe that of the star instead. 848. The Computation — (1.) Having reduced the alts, to the time of the mean of the distances. No. 6(50, find the Gr. Date. At sea, the Gr. Date is required only to the nearest hour ; but if the moon's alt. is not observed, it must be found with precision. Reduce the hor. par., and thence the semid., from Table 40. Augment the semid.. Table 42. For precision, correct the hor. par. by Table 41. (2.) Find the App. Alts, of the centres by applying the ind. corr., dip, and semid. Correct the observed distance for ind. error, and add the semi- diameters of the bodies: the result is the apparent distance. (3.) Find the Sun's Corr. of Alt. by subtracting the par. in alt. from the refraction. Find the Moon's Cor. of Alt. by Table 3S). Correct for the therm, and barom. whenever these instruments ai-p accessible. Tables 32, 33. (4.) Find the true distance by No. 842, 844; or, for precision, No. 84.5, and apply the corrections, Nos. 852 and 853. (5.) For the G'M.T. Find, in the Nautical Almanac, the two distances between which the true distance falls. Take out the first nf these, and set it down under the true dist., and write against it * The reason of this order, as a general rule in such rases, is, that the outer body prc- fenes uniformity in its change of alt. for a longer time than the other, and consequently its alt. may lie reduced, by simple proportion, to au ii.tcrmediate time, with less error than the ill, of the other body. See No. bM. FINntNi; THE LONGITUDE. 503 Its prop. ]o'^ f^ivcii ill the Nautical Almanac ; note also tlie tiiii« (that is, the tiiree hours) corresponding. Take the difference between the two distances thus set down, and from its pro|). log. subtract the prop. log. taken from tiie Nautical Almanac; tiie remainder is the proj). log. of a jjoition of time to l)e added to the time from the Nautical Almanac. Tlie result is the G.M.T. of the true distance. For jirecision, see No. 856. The G.M.T. being found, the long, is determined. Ei. 1. 11 M S. Eden, April ytli, 1831, Int. by ace. 34° 30' S., long. 42" W., walcli slon on tlie cliron. 8'' 16"' 31"; chron. slow of G.M.T. 4'' 54'" 33'; lieight of eye, 16 feet ; iDd forr. -7' 36" ; had the following observations : required tlie error of the chronometer. Thneu by Waich. Alt. "J AIlQ Dintance. ■i'' 57"'24' 39"*' •i 5« 36 47° 33' I I 29 66 o' 8" (the mean of tlirce sights., > 1 47 47 5i I 8 18 36 42 Reduction of tlie Altitudes to the time i'' 1" 29». J 39' 2', 12I' ST" 24*. 4" 3*' 41 . 2 20 18, 54 1-644 ® 47° 33'. o''5 8'"36', 1 19 8-7S2 47 5^. > 5 47. -5^ VJ ? Moon's Alt. 38 10 Redjced Obs.-Time, il" i"' Time by Watch Watch slow of Cliron. "■79b 8601 o'976 i-37» Sun's Alt. 47 41 Alt. J 38° lo'j Alt. Q47°4i'; Obs. Dist. 66°o'S". J H. P. on the 7th, nonn, 56' 34'' midnt. 56 59 Var. in i2i> T; Prop. Part for 2'', 4", H. P. 5638 Corresp. Sem. 15' 26", aug. do. 15 35". Gr. Date,* 7th Obs. 1 38° 10' Ind. Corr. — S' \ Dip - 4 Sem. — 16_|_— 28 jApp. Alt. 37 42 J App.Alt. 37° 40', H 2 pts. - 2- 1 38 +30 I +28 J Corr. of Alt. 43 33 Clearing the Distance (No. 8(2), to the End. Obs. Q Ind. Corr. Dip Ai>p. Alt. 47 56'. 43' 5" 7" 4'' Obs. Dist. 66" 0' 8 Ind. Corr. - 7' ^6" ) ]. Sem. +.5 35 + 4 © Sem +15 S9 ) +^3 ■;» 7 4S App. Dist. 66 24 6 © A pp. Alt. 48", Refr. 53" Par. n Alt. -6 Corr. of Alt. 47 J Alt. 37*/. »n<i Co-dist. 23}" Alt. 47 'and23°, M 159 3. N4';- M ,37-5, N 33-4 37' and 159 Dep. 9V _(gr., 6) 47t' and 137-5 D.-p. 10.-4 (i;r.) Corr. + 0^ 24 50- (DifT.) 680 J 2 Corrs -0 29 32 jl-tCorr. -29' 37" -0 29 8 J 2d Corr. 44', X" ) . .„ 30, 3 j ^""- ^ = A. Dist. 66 24 6 True Dist. 65 54 5K Do. at oi' 67 8 p. log. •305. ~ T"^ p. I<,g. •4412 jh.in,,.. p. log. •136, T. ofPr. Dist. J3.M.T._ »_'i_35_ il pcrlorm 1 310 NAUTICATi ASTRONOIIT. Tl.e watch being slow of the chroii. 8'' 16™ 31% the time of the obs. by th« diion. » 1™ 29' + 8'' iB"" 31', or 9'' 18" o'; the chron. is therefore f' 6" 26' fast, or 4" 53- 35' w of the G.M T. Now, hy Table 58, an error of i in the dist. causes, in tliis case, an 01 of 2" 8" in tlie G.M.T. ; hence the result may be considered as confirming the error tlie chron. nearly enough.* Ex. 2. Sept. 28th, 1878, at s' Ii» 40* r. 146° 55' W. ; obtained the mean of 7 distan< ij^22° 37', obs. alt. J. 18° 7'; height of eye ship ; in lat. 48' 50' N., long, aco sun and moon 34° 48' 16", obs. alt Gr . Date, Sept. 28'' 12'' 59"" 20' J's Red. H.P. 60' 35" Aug. Scmid. 16 37 ©•s alt. 22^49' Kef. 2' 18" corr. of alt. 2 10 App. Dist. ©■s .\pp. .Mt. i's App. Alt. 35° 20 54' 22 49 18 20 8° 20' H.r. 60 54' 4 »'s Corr. of All. 35' _tL3 54 37 Clearing the Distance by No. 844. (.^ .\pp. Alt. 22^49 COS. 99646 3 App. Alt. 18 20 COS. 9'9774 App. Dist. 35 21 sin. 97624-1 ^6^3° 9-6990 J. 9'5664 Half Sum 38 15 see. oio5oJ 1st Rem. 15 26 cosec.o-5749 ad Rim. 19 55 cosec. 04677 Corr. Alt. 54' 37" P.L. o 5179 ' 2' 10" I'.I-. 1-9195 1st Corr. 41 34 r.L. 0-6366 ;i Cor.2 10 r.L.i-9iS2 Dili-. "1^3 Dist. 35° 55' Corr. 38" 13 2 2d Corr. + 36 35" .56 47 App. Dist. lCor.Alt.54'37"l 3d Coir. 2 10 / 34 »4 7 1st Corr. t 41 34 0's Corr. Alt. +210 2d Corr. + 36 At 12'' RI.T.G. M.T.S. 35 8 2-/ „, 34 34 33 P-L.24I7 o 33 54 l'-L.71i' o"59"' 8'1M..4834 2S''i2 59 8 ^ 8" 3 n 4° 9 47 28 = 146° 52 7i. r.x. 3. Sept. 1st, 1878, at 4'' 40'" 4' r. 5' \V., obs. alt. Q 20-^ o', obs. alt J^ 62^ f 2' N., long. acc. 6"; height of eye 18 Gr. Date, .Sept. i'' 4'' 44" 24' J's Red. 11 P. 59' 38" Aug. Semid. 16 31 ©'s Alt 20 12' Ret. 2' ; ©'• Corr. of Alt. Tl '58' 51" App. Dist. ©'s App Alt. ys App. Alt. 6i 42 62° 40' II. I'. 59' 26 36" 38" --8/-^''* .i's Corr. of Alt. 1(5 52 • The Nautical Alm.iiiat, before 1834, was c rwili is therefore Grt«iiu'ieli app. time. This do • elue ol u irvre etunplc. iiputed for FINDING THE LONCilTrDE. 31] 0App All. iO"Ii ) App. Alt. 62 42 COS. 9-6615 Api". Hist, fci 59 sin. 9-94591 144 53 96990 ^ Half Slim 72 26i sec. o'5204 ) 1st Item. 52 14$ cosec. 0*1020 M Hein. 9 44^ Ci.rr. Alt. 26' 52" PL. 0-8261 2' l.t Corr 31 39 l'.I..o7549;KV.r. Uiir. 4 4' Disi 62° 27 Con-. 4" 5 _° '2d Coir. +4 ■■ 9'97M 01653 SCI-. 0-7716 M,. I «6o2 l.y No. 814. Api l8"lM,.2-769: ..\lt.26'52'| :!(! Corr. o 18 J 6 l!=t Corr. O's Corr. Alt. 0,1 Corr. True Hist. 6 At 3" 6 -27 10 y 2 29 H o 4 5 55 7 3Si r>. 58 .5 p. "44"'33"1'. IM.T.G. M.T.S. Loi Kx. 4. Sept. 3otli, 1S78, at 4'' -■4'" 46" I'.m , IM.P. at : ' 40' W.; obs. ali.Q 16° 12'; oils. alt. J^73^ 14'; "I's. Hi ip. l.it. 17° 9' S, lonj;. a . 60^" 22' 59'' ; lieiglit ol" I G.M.T. Sept. 30'' 1 1» 35"' 26". corr. H.P. 58' 59", aiij; .semid. 16' 21". Q's app. alt. 16" 24'. J's app alt. 73' 26', app. ilist. 60° 55' 2i"; 0's corr. of alt. 3' 8", J's eurr. of alt. 16' 32", true dist. 61° 10' 36'. Long. 102'' 47' 30" W. 3. Lunar Observation hy a Star or a Planet. 849. 77ie Observation.— Take the alts, ns directed, No. 847. In taking tlie distance, direct the view to the star, make the contact nearly hetween the star and the illuminated edge of the moon, wliether it lie the nearest or fartiiest limb; clamp the index, put in the telescope previously adjusted to distinct vision by the star, and com- plete the contact by bisecting or splitting the star upon the moon's When the moon is bright, it is necessary to use a shade. The setting of the inde.x, No. 847 (2), is ,i more important step in observing with the star than with the sun, for the amount of distance i.s often the only security for employing the right star. For jn-ecision, note the azimuth as directed No. 847 (.3). 850. T/ie Computation. (1.) Proceed by No. 848 (1). Fur a phiiiet, take out the hor. par. from the Nautical Almanac, and reduce it. (2.) Find the ajip. alts, as in No. 848 (2). l"'(ir the ap|i. (list., correct the observed dist. for ind. error. When the iiKircKt limi) is observed, add the moon's seiuid.; when the furthest, tuUtract it. * It lias been recommended to observe the star open of the moon's edge, leavini; a dark •pace of about 40'. But this dark space will appear differently in different telescopes ; and, iiiorenvcr, it is better to be in the practice of observing accurately than loosely. The inaccuracy which arises in bisecting a |ilanet when it is not, as we should say of the iniion, at the full, is but small ; since, even in the case of Venus, the only plani-t which ever appears a» a crescent whin observed with the ninoii, it i-.iii scarcely e.^cced 6' or 8'. It hu been propoaed to correct for this hy a special coiiipulatiou. 312 NAUTICAL ASTRONOMY. (^3.) Find the star's corr. of alt., which is the refraction. For a planet, apply the par. in alt. from Table 45. For the moon, take her corr. out of Table 39. For precision, correct for the height ot the barom. and therm. (4.) Find the true distance, and proceed as in No. 848 (4), to the end. Ex. July i6th, 1826, near midnight, lat. by ace. 27° 5' N., at 2>' 34"' 15" by She cnron., obs. alt. jl 35° 12'j obs. alt. Fomnlhaut, 12° 51'; obs. dist. farthest limb, 70° 1' 10". Ind. corr. -20" ; height of eye, 16 feet : required the error of the chron. supposed fast oa o.M.T. I"" 6<" 7.5». Time by chron. z^ 34"' I H.P- i6th. midnight 59' 41' Chron. fast. I 6 | 17th, noon 59 35 Var. in 12" 7 Red. H.P. 59 4' Corresp Sem. 16' 16"; Aug. do. 16' 27'. G. D. 6th, past midnt. 1 2« (.bs. Alt. » 35° >^' Hip -A Obs. Alt. * 12° 51' Obs. Dist. 70° i' 10' D.p —4 Ind. Corr. — 20 Sem. +.6f * A. Alt. 12 47 J Sem. -.6 27 J A. Alt. 35 24 A. Dist. 69 44 2J 35' 20', and H.P. 59'. 46' 47- * Alt 12° 47' 4. - ^'\ 4< , +33 I + 31 J Corr. of Alt. 47 .8 * Co r. of Alt. 4 >»' Clearing the Distance (by No. 812) to the End. 1 351° and 20°, M 130-7 N. 26-0 + 13" and 131 Dep. 29-5 (gr.) 3-5 5 1st Corr. —I 36" J 2d Corr. {'^'([^^ ] +6 ' and :o°, M 109 J 35° * Corr. J 2 Corrs. and 109 r -0 1 30 )ep. 62-5 54- > p. log. p. log. p. log. (gi; A. Dist. True Dist. Dist. at Mid. G.M.T. T. by Chron. + 42 69 44 ^3 69 45 5 70 3. .5 46 10 ,•- 26-55' 2 34 >3 •1747 •5909 •3162 Error, /a»7 I 7 i» Ex. 2. Sept. 7tli, 1838, P.M., lat. 3° 2' N., long. 4'' o"' W., at 12'' 57°' 8" by watch, obs. five distances of themoon's nearest limb from ,\ldebaran, 27°47' 12". App. alt. !(: 26° 32'; app. alt. 3> S3°34'; watch slow 9'" 17" of M.T. j ind. corr. —1' 10" : required the longitude. D red. H. P. 59' 48' ; true dist., by No. 815, 28° 37' 17" ; dist. at XV'', 29° 47' 47'. Long. 59" 56' W. Ex. 3. Sept. 2cl, 1840, P.M., lat. 3° 2' N., long. 60° o' W., at 8'' 48"' 39" by watch, obtamed the mean of 5 distances between Saturn and the moon's nearest limb, 89" 41' 55° j ind. corr. —1' 25"; watch slow of M.T. 7'" 33"; app. alt. J 53° 3'; app. alt. Sat. 23° 34'. Ex. 4. July I4tli, 1878, at 2' 10"' o" a.m., M.T. at sliii), lat. ti" o S., U«p. ;irc. 14.9' 30' E., obs. alt. Aiitaits 19" 33', olis. alt. _}) 51^ 48', obs. di»t. nuar limb 7^ xz' 49', lifight of eye 314 feet nxDixr THE LO^'on•UI»K 315 ", err. 11 .list. 79° r. 56' 4 38' .s". l". a.icc. sumul. * err. 2 44", J's c ", + ap orr. 34' >. alt. :9°28', 9", truf ilist. • 149° 33' !•:• at V s. alt. •JO"' A.M ; M .l5^-°54:.' T. atsliii), lat. ao^ 10' )s. .li^t. near li.ub 86° N.. loiiR 45' 44" aec.75°W. l.uiglUofc>-. r, M.r. .i.iiy 13' 4' 12- «'.s ai.p. Ul. 51^ 5./ app. 79' 29' 4''- Kx. .'i. June I9tli, 1S78, iil)s. alt. Vciuis 23'" 14', ob 16 tlvl. vJ.M.T. June 18' 21'' 30"", corr. H V. 55' 7", aii^'. seiiiid. i 5' 14", Venus' app. alt. 23° 10', J's app all. 53° 5' J app. dist. 87^ o' 58", Venn-.' corr. »' 9", J's corr. 32' 23". true <list, 86° 43' 48". LoN.J. 75° 6' I5"W 4. Hjiccial Corrections. 851. When precision is required, it is neccssMry, Iwsides rem()\inj» from the distance the general eHects of refraction and paraiia.x, to apply certain corrections. [1.] Correction for the Elliptical Figure oftht Disc. «o2. Since tlie refraction of each point of the di.sc of the snn or moon is greater as the alt. of such point is less, and since the change of refraction is pi'oportional to small changes of alt., the n|)])er and lower halves of the circular disc take more or less the figures of ellipses, the lower half being more flattened than the ujjper half. The distance, therefore, between the centre and the limb, as it would actually be observed, is less than the horizontal semidiameter nf the Tables. The elliptical figure of the sun, due to this cause, is often cons])icuous at rising and setting. The correction in Table 53 is to be subtracted from the semidiameter.* [2.] Correc/im fijr the Spheroidal Figure of the Earth. 853. The true distance found from the data, as above, is deduced on the supposition that the earth is a sphere, instead of a s])heroid. 'J'lie true distance found is, in fact, that corresponding to a s])bere of smaller dimensions than those circumscribed by the cqnator,t and to an horizon differently placed with respect to the equator, or to another latitude than that of the spectator. Since, however, the mere change of the ])lace of the spectator would cause no alteration in the apparent angular distance of two stars, the change of distance arises solely from the variation of tlio apparent place of the moon, produced by the changing of the observer's astronomical latitude for the geocentric latitude. The change of place of the moon is thus in general the resultant of a change both of her altitude and her azinuith. This correction is at the equator and ])oles, and is greatest in lat. 45°. As it cannot nuich exceed -^f^ of tiie reduction of latitude, it may in practice be omitted, but the ellect rarely disappears altogether. * We have not applied this correction, because at low altitudes, the only case in which It is sensible, the observation is not to be depended upon within such small quantities. t The correction on this accouut haE already been made in Die reduction of the moon'i °f|uatorial parallax. 311 NAUTICAL ASTRONOMY. 854. To coiToct tlio distance. Enter Table 55 with tlie lat. and tlie alt. 90°, and take out tlif number. For Part I. Enter Table 5 with the complenicuts of the moon's aznnutli and of tlie angle at the moon (fonud by No. 842 or 844),* Hiid take out M. Divide the number by M. For Part II. Enter Table 5 with tlie moon's aziinntli and the angle at the moon, and take out M. Divide the number by M. The quotients are in seconds, and are to be applied to the distance as follows. Part I. Part II. J to the E.twa.. In N. Lat. In S. Lat. Angle at the lesn than 90" Azimuth "f >'"■ ■» ' Suno to tlie riyhl rStar to the left Suno to the right rStar to the loft less than 90° greater than 90" sub. add add sub. sub. add J to the Westward add sub. sub. add Angle greater than 90° add sub. Lat. 48° N. ; moon's alt. 30*^; star's alt. 61°; dist. 54°; moon's azim. S. 72^ E. j Co-az. 18°, Co-ang. 56°, — -;— = 6", mbtractxve. Az. ■^^°, Ang. 34°, = 3", stiblractwe. 390 Hence the Correction is 85.5. When the moon is near the zenith, or when her alt. exceeds 80*?, with the lat. and the eompl. of the star's azimutii as an altitude, take out the seconds from Table 57, and divide then^ by 100; the quotient is the correction required in seconds. AVhen the star's azim. (reckoned as above) is less than !)!)°. subtract the corr., otherwise add it. * Since the angle at one or both bodies, which is given by the method No. Xf>, neressary in making the corrections, No. 852, 853, and since that method affords butli ; apiiroximation hf which the long, by ace, if greatly in error may be corrected, ami at t uanie time a check against any important error in the rigorous process itself, it will advisable to employ it on all occasions. The angle at the body may be found from No. 844, when 'hat method is employe thus: — Take the sum of the logs., rejecting the const. 9'C990 ana the prop, log.; the ar. c log. of this sum is the log. sine squaie of the angle required. En. No. 844. Sum of four logs. A HOLE at J) 55' 30 0-6641 *<!• 9'33S9 Sum of logs. Anole at ® 77 riNPiNo Tiir, i.oNr.iTunr,. 315 13. ] Correctwn for the rnerjualih/ of the Moons Motion. Wo6. Since the moon does not generally cliangc her Jistanrp f'roiri the sun or a star at the same rate, both at tlie heginnini:: ami end of S hours, it is often proper to apply a correction to the (ir. INI.T. found, which, iu the extreme case, may he in error oO' of long. When the distance exceeds 26'', this correction will not exceed I5'of long. ; when the distance is near 90°, it will not exceed '2'. !n general, it is smallest in the case in which the sun or star is in a direction perpendicular to the line of cusps or horns. 857. Take the diff". between the ])rop. logs, in the Nautical Almanac against the two distances between which the given true (list, falls. With this diff., and the portion of time found in No. 848 (5), enter Table 57, and take out the seconds. When the prop, logs, in the Nautical Almanac are increasinrj, siihtrnct these seconds ; when decreasiivj, add them; the result is the M. T. at Greenwich, Corrected. Ex. 1. Dist. in Naut. Aim., preceding given dist., ZZ-'SS' 2i" prop. log. 3079 following do. 24 56 56 do. 3054 {(hcrcaninij) Diff. 25 under Int. 13"', [add) Diff. 25 Corrected G. M. T. o 26 E.t. 2. In Ex. 2, No. 850, dist. 29" 47' 47" has the prop. log. 2527 ; the next in order has 2581 ; the ditf. 54 gives 14' to be subtracted ; and the long, corrected, 59^ 52' W. 858. l^egree of Dependnnce. The true distance is affected by errors of observation, and by errors of computation. An error in the distance, of whatever kind, produces, on the average, about 'M) times its amount in the longitude; thus, 10" error of distance ]iroduce about 300" or 5' error of longitude. The observed distance is liable to the ordinary errors of angular ilistance, the chief of which are, perhaps, most usually that due to defect of parallelism of the telescope, and that arising from making the contact above or below the centre of the field. Irradiation is also included in the errors of observation. 8.59. The error of the computed result arises from two sources ; the errors in the elements of the observation, and those of the method of solution. (I.) Under the first of these heads are comprised the error.* in the horiz. par. in reducing it to the Gr. Date, and for the figure of the earth, the error of the tabular semidiameter ;* and that of refraction in low altitudes. (2.) The effects of errors of a few minutes itt the altitude are iii- hi'iisible. Hence an ill-definud horizon is no great detriuieut to a ♦ The Grcenvriili obs.-rvations shew that the semidiameter of the moon, w given is Bui ;khardl's tables, is 3" too small. — See " Green. Obs." 1837. 31(5 NAUTJCAL ASTRONOMY. good observation; and hence, also, in computing the Altitudes, pre- cision is not essential. This last remark is worth attention, since the calculation of altitudes is a heavy addition to the worlc of a lunar. On the same account it will not be necessary to consider the change of place during the observation, unless the second alt. of either body be lost. (3.) The importance of correcting for the barometer and ther- mometer has Iwen noticed. No. 838. The atmospherical correeticm is of most consequence at low altitudes, and when the bodies are in or near a vertical plane. (4.) The smaller corrections, namely, reduction of equatorial pa- rallax, corrections for elliptical disc, for the figure of the earth, and for unequal motion, cannot all be applied the same way in atiy observation ; compensation will accordingly take place to a consi- derable extent even when these corrections are omitted altogether. It will, however, be advisable to apply the latter correction, No. 856, when large. 860. The error of the method of solution, No. 842, may be estimated for distances exceeding 50° at not niore than 20*', in general, or 10' of long. Method II., No. 844, will, in the same cases, be more accurate. 861. The eflects of errors in general, and especially cou.slaut errors of observation, are removed in a considerable degree by ob- serving ei/tinl distances on opposite sides of the moon, since the errors of the resulting longitudes will be of opposite kinds. The true long, will not, however, be the mean of the two erroneous longi- tudes, unless the moon changes her distance from both bodies at the same rate. When the two longitudes in such a ca.se differ widely, add the pro]i. log. of their difference in time to the prop. log. of the yrealer motion in 3 hours (which is the smaller of the prop. logs, in tlie Nautical Almanac), and tlie ar. co. pi'op. log. of the sum of tlic two 3-hourly motions; the sum is the prop. log. of a portion ofiiio lime. to be ajiplied to the long, obtained by the star whose prop. log. is employeil. Since the true long, must fall between the two given results, it will be known at once whether to add or subtract. When the sum exceeds 3°, read the degr. and min. as iiiiu. and sec. Er. The long, by Regulus, in a certain case of a lunar, is i*" 37"' 15"; by Antarc?, I" 40"' 58" ; the distances being nearly equal on opposite sides, and observed by the same ohtierver with the same instrument. Tlie 3-liouriy motion of Regains is 1^ 45' 31", that of Antarcs 1' 30' 19": required the True Long. Long. Reg. 2''37"'i5" 3-liour. mot. i°45'3i" p. 1. (\' ^■-,") z'oioi Ant. z 40 58 do. I 30 II) 3 16 o ar. CO. (3' 16") 8-2c;S8 I 5 (9' more than the mean). FlNDIXfi THE I.ONOITUnK. 5)17 862. Aflor the result lia.-. liceii olitaliied with tlie ufjiost care, lliore ri'iiiaiiis the error of tlic liiiiai- lalth's, wliich aiipeai's to l)e ahoiit U''5 of R.A., or 4' of long. 'J'his can be removed only l>y careful exaniination of observations: of the moon, made near the same time in a fixed observatory. In general, the result will have more value as the jnoon's horizontal j)arallax is greater, because her motion is then more rapid ; on the contrary, the result is of less value as the horiz. par. is less. Since the changes of the moon's It.A., at their maximum and niinimuni, are nearly in the ratio of 5 to :{ and since the change of R.A. is in a considerable degree, though not in exact proportion, greater as her distance from the eaith is less, it is evident that the place of the moon at the time of i>bser\ation materially affects the value of the result.* 5. Computation of the Altitudes. 86:?. When the altitudes are not observed they must be calcu- lated. iM. T. is supposed to be given. (1.) Reduce to the Gr. Date tlie sid. time at mean noon, also the U.A. and decl. of each body, unless one of them is the sun, in which case reduce the e([uat. of time instead of his R.A. (2.) Find the hour-angles, Nos. 609 to 612, and compute the alt. of each body. No. 667. See No. 86<J (2). For the apparent altitudes. Take out the corrections of altitude to the true alts., found as if for app. alts., to the nearest minute, and apply these corrections the contrary way to that ihrected in the rules, Nos.' 644, &c.t Ex. Sept nth, 1838, A.M., at Fort St. Joaquim, lat. 3° 2' N., ' 49"' 40" by watch, obtained the mean of five distances of the sun and id. corr. — 55"; watch fast of M.T. 3'" 2"; therm. 85°; barom. 297 long. 4'' T. by watct gh 49-" 40- J H.P. lit 1, noon 56' 56" Watch fast 9 4'' ^8 midn. 24 M.T. 2, 46 ^8 5 Long W. 400 56 56 Gr. Date, cth. 2? 46 38 Red. H.P. 5ft S3 or iih, . 46 38 Sen.id. 15 30 * In combining the results of different observations for tlu Inncitude of a place, regard would be had to this and other cin mn weight to each several result. The final determination of p.i^ii observations made at different times and under different circiiinsi grapher or geographer rather than the seaman or traveller, and t As tlie altitudes in a I are necessary to remove the tmplnycd. It will be pnident to verify the Table .')'), in order to avoid entailing comiiutation. are not required with precision, Tables A^ anil 44, which uracy of using the true alts, as arguments, will rarely be by the method of inspection (see Expl. of i.terial error on the whole of the subsequent 318 NAUTICAL ASTRONOMY. c-nts for computing the Altitudes.' Sid. T. noon ii''jo"i4' M.T. 21" 46'" 38' J R.A. nth, l^ 5''4''M l\ lo* 1 + 3 24 2. 5 43 4» 4f>"'5, 7 ♦ 17 2. 50 2 w. 2 28 Ked. S.T. II 20 31 © H.-ang. 2 9 58 M.T. 21 4.6 38 © Decl. 33 7 9 t \ 45 N. 46 38 .>-5866 Kq.ol-T. i.th, 3-23. 22 53 oh i«'55' 19727 12th, 3 44 5 41 '4 21 4 3? 'iS Red. R. A 5 43 9 Rb.l. Eq T. 3 24 Red. De,l. 4 36 58 N. 33 7 9 P. Dist. 85 23 J H.-A. 3 24 1' De<:l. 28° 36' 48' N. U ■3" J Decl. 28 38 P. Dist. 61 22 Computation of the Altitudes. © H.-A. 2" 9'%8' 5 H.-A. 3I' 24™ o« .Suppl. 9 50 2 tin. sq. 9-96461 9-99859 Suppl. 8 36 sin. sq. 9-91098 P.O. 85° 2 3' P. D. 61° 22' sine 9-94335 Col. 86 58 172 21 sine 9-99939 Col. 86 5S 14S 20 sine 9'999^9 Arc X- 146 36 sin. sq. 9-96259 Arc X 1,5 21 sm. sq. 9-85371 3'8 57 263 4. ^5 45 32 59 159 28 Bine 9-54500 131 50 sine 9-87221 12 53 sine 9-34824 .6 29 sine 9-4';29i 32° 29' sin. sq. 8-89324 54° 45' sin.sq. 9-3^';>i © Tr. Alt 57 3' jTr. Alt. 35 15 Corr. Alt. + I Corr. Alt. - 46 82° 16' 5. e A. Alt. 57 32 J A. Alt. 34 29 -55 82 ,5 56 + 15 55 -H 15 50 + 9 A. Dist. 82 47 5a 57° 3^', 37" J 34° ^0, H. P. 5 6', 44' 50" zJ. 9 — 4" 85°, -2" 3» 53 " -^43 + 39 2, 45 29 © Corr. of Alt. 30 85°, — 5 \ — I ] -^ 6 J Corr. of Mt. 45 35 Proceeding to clear the distance by No. 845, the log. diff. is 9-996092, and the true dist. 82°4'5i". The next dist. preceding is 82" 58' 33", at noon ; and the G. M.T. i''47"'o', or Long. 60° 5' 30" W.f * To adapt this form for computing the altitudes to the case of a planet, put the plmet'i hor. par. in the place of the equat. of time ; and in the next column the planet's K. A. t Tliis observation, and those in Examples 2 and 3 of No. 800, wfxe taken, with 9e»ei»] other*, by Sir Robert Schomburgh, to whuiii I am indebted for them. F«NDI\(; THE I.ONGITUDK. ."19 Til. Bv THE Moon's Ai.tmude. 8()4. Since Mean Time is deterniinefl i)y the lionr-aiigle and }\.\. o/ a celestial body, the U.A. may be deteriniiied from tjie M.T. rinii the hour-aiiglo, the latter being eom|nited from the observed aUittide. Now the moon's R.A. being- given in the Nautical Alma- nac for certain points of time, the time at Greenwicii corresponding to any given R.A. of the moon may be at once found. 'Jiie moon's altitude has accordingly been often tiius employed in determining the longitude ; but the method requires much caution, because an error of altitude produces, in the hour-angle coniputetl fiom it, a quantity greater than itself, except in the single case in >vliicii the oliserver is on the equator and the body on the prime vertical, when these errors are equal. Accordingly, since an error in the moon's hour-angle appears in its full amount in her deduced K.A., and since the R.A. changes at the rate of about 2"" only in ai. hour, the longitude required is vitiated to the extent of not much less than thirty times the error of altitude in the most favourable cases. It is evident, therefore, since the place of the sea-horizon is often doubtful from 1' to 3', that the result of a sim])le lunar altitude must be in general greatly inferior to that of a lunar distance, in wiiich a good observer rarely makes an error exceeding half a minute. But as many persons, who are not sufficiently expert in ihe lunar observation to obtain on all occasions a satisfactory lon- gitu<ie, are nevertheless capable of observing altitudes with precision, Hud, moreover, as tiie stars, when the air is not very clear, are often li)o fivint for the lunar observation, the former method may, on some occasions, prove of service, provided tliat proper stei)s are taken to diminish the ettects of the errors of latitude and altitude. Since on the equator, when the body is E. or W., an error of 1' in alt. jiroduces an error of 4* in the hour-angle, and an error of 8' in lat. 6U° (or in the ratio of the secant of the latitude to 1), the method serves better in low than in high latituiles. If the resulting longitude ditfers much from the long, by account, the computation should of course be repeated. 865. Limits. The azimuth is the same as that laid down for determining the time by a sinn:le altitude. No. 778. The alt. should in general not be less than 6® or 8° ; and when the barometer anil thermometer are not at hand, not less than 25° or 30®, especially in \ery cold or very hot weather. 8t)tt. 'I'/ie Ohsrrvatinn. Observe the moon's alt., noting the time. If the mean time is not accurately known, obtain observations for it. ^.t sea, the uncertainty of the apparent dip may be removed by ••t'ferring the moon's altitude to the opposite point of the horizon, an wi'll a^ to that under iier (No. 5-'35). 320 NAUTICAL ASTRONOMY. liiit it will be preferable to observe tiie diffeirnce of alt. of the moon and some star on nearly tlie same bearing, and to a.]>]Ay it to tbe star's alt. found by computation ; for the time may sometimes be more nearly known than the lat., and the alt. of a star computed more nearly than it can be observed. For Ex. Suppose, in lat. 40°, the J bearing E.S.E. (true), that the place ot the sea horizon is i' 30" in error, and the tune in error 5'. Then the error of the H 's computeil hour-angle (and therefore of !ier R.A.) will be 9** (No. 671), and the resulting error of long, about 4'" 30', or i" 4' (Nos. 858, 864). Now the error of the computed alt. of a star E. or W. due to an error of 5' will here be 56" (No. 671) ; hence the error of the long., as deter- mined by the moon's alt. referred to this star, will be diminished in the proportion of 1' 30" to 56", that is, from 64' to 40'. 867. 2V/e Computation. (1.) Find the Gr. Date, and reduce to it the Sid. T. at mean noon, the moon's decl., and thence her \\6\. dist., her hor. par., and semidiameter ; correct the hor. par. by Table 41. (2.) Add the M.T. to the red. Sid. T. ; the sum (rejecting 21'' if it e.xeeed 24'') is the R.A. of the meridian. (:3 ) Correct the alt.* (4.) Compute the moon's hour-angle, No. 614. (5.) When the moon is to the E. of the meridian, adil her hour- angle to the R.A. of the mer. If the sum exceed 24'', reject 24''. VViien to the W., subtract tbe hour-angle from the R.A. of the nier., increased, if necessary, by 24'' : the result is the moon's II. A. (6.) For the G. M. Time. Set down in order this R A., that preee<ling it, and that following it (from the Nautical Almanac) ; take the ditt". between the 1st and 2d, and between the 2d and 3d, adding 24'', if necessary, to effect the subtraction. To the constant 0-4771 add tbe prop. log. of the first of the ditfs. and the ar. co. prop. log. of the 2d ; the sum is the prop. log. of a portion of time to be added to the hour at Green, of the middle one of the three right ascensions: the sum is the G. M.T. Ex. l.t Jan. 5th, 1839, lat. 4° 54' obs. alt. 'J 30° 6' 20" to the W. ; ind. corr. —35"; height 1 barom. 30-0 inches : required the longitude. M.T. 3 3".3'W. 20'' 56".4i' 12 52 3) Decl. 5th,at23'>,o°i6'39"S. Dift'. for icw, 142" o°,6 39" 140", 9™, 2' 6" ) 2 %^ l-l] 2 ,6 Red. Decl. 18 55 S. Pol. Dist. 89 41 5 H.P. 5th, Mid. 6th, Noon Var. in I2i> 7"-2.ind iiJS Equat H.P. Corr. Table 41 C<irr. .Semid. Au-m. Aug. Sem. 54 25 ■■2 54 18 -o Gr. D. 23 9 33 7 '2 Sid. T. 5th, 23" 9m 33' 18 57 34'8 3 46-7 ■•5 6 -S 54 25 -2 54 .8-4 Red. S. T. M. T. '9 56 40-8 .4 48 7 R. A. Mer. •5 58 3-9 '4 55 * It cannot be worth while to follow the 2d and 3d precepts of No. 655, unless the observation is in every respect such as to afford extreme precision in the result. t These examples are selected from observations made by Mr. J. C. Bowring on board ri. M. S. Stag, with which 1 have been favoureU oy Mr. I'entlsnd, her JIajesty's lute consul- general at Bolivia. FINDING TIIR I.ONCITI'DE. 321 Ob». Alt. J IJip. -3 '-■= ) «9" 40' and 54' 7 »«* 5' i8'a</<ii6 App. Alt 45' '4 + '3 Th. 82°a</</ 6 Kar.3. 45 i-' 6 Alt. 10° 33' 3" Ut. 4 54 " P. Di»t. i!, 4. ; 125 S 8 joc. 0-001590 cc«ec. o-oocio; 62 34 4 COS. 9-663417 32 I I sin. 9-714415 Hour-ang. 31' 57"' 2 5-2 «in. «q. 9-389429 0" 8-30-3 '"3»56 ^3 G.M.T. 23 8 30 M.T. 20 56 40 L0.VO. 2 11 49 1 5 or 32° 57' W. An error of i» of R. A. would produce here 34* or X j' error of long., as the R. 1. change* very slowly. An error of 1 of alt. would cause 4' of R. A, and 34' of long., and an error of 1' of lat. only C-i of R. A. The moon's axim. U 87"". Ex. 2. Jan. 2 obs. alt. 7 42 25' Longitude. (Jr. Date, 23d 6'' Red. S. T 20 K.A..\ler. 25 long, by ace. 42° 39' W., at 3' 32"" lo' M.T. r xy" ; height of eve, 12 feet 1 required the J Corr. of AlU J True Alt. An error of JuccB 4*'3 error 5' of long. of R. A. produces here 25", or 6' t'R.A., or 27' of long. ; and an .5' X. 45 Equal. H. P. Red. do. Corresp. Sein Aug. Sem. 58' 41"- • 58 40 -8 16 16 11 lour-mgle 3'' 23"'27-o > R.A. 35 '^-7 )o. at 61' 3 4 20 -1 3 6 41 -8 fi.M.T LO.NG . 6" 22" 1 5- . 2 so 5 or 42" 3.' w or of Ion or of 1' '. ; an error of f lat. causes c» 'of alt. pro. 9 of R. A., ur 868. Wlien two or three observations are taken on tlie same side of the meridian and prime vertical, the true long, is not the mean of the results, but is nearer to that which is furthest from the meridian. When two observations are taken on opposite sides of tlio nieri. dian and on the same side of the prime vertical, tiie right ascensions rorjlting will be affected in different ways by the same errors of altitude and latitude, and the true long, will be between the two results. 869. Degree of Dependance. This is determined by the effects produced on the hour-angle by given errors in the alt., lat., and pol. dist.. No. 615. It is evident, from the remarks above, that unless considerable care, and some skill, are devoted to diminishing, accord- ing to the circumstances of the case, the effects of errors of latitude and altitude, if cannot be prudent, notwithstanding the occasional »uceess of observations of tiiis kind, to depend upon the result r3 nearer than '\ of a Af^irfi'. 322 NAUTICAL ASTRONOMT. Od shore, when the lat. and time are accurately known lbs reeult may, with proper attention, he more satisfactory. No. 862 applies to this observation. IV. By AN OCCULTATION. 870. Tjie moon in lier perpetual revolution round the earth necessarily passes over every star or other body in her path at certain periods. Tiie disappearance of a star or planet, called the immersion, and the reappearance from behind the body of the moon, called the emersio7>, being instantaneous, the phenomenon affords tlie means of determining the longitude at all places where it is visible. At the instant of occultation the apparent R. A. of the moon's limb is the same as the R.A. of the star ; the effect of the paralla.x of the moon being removed by computation, the true R.A. is de- duced, and the G.M.T. thence found. 871. This observation affords, in favourable cases, the most decisive results, because it is both instantaneous and altogether independent of instrumental adjustments. On board ship the motion prevents the telescope, v/hich is almost always necessary, tiom being kept steadily directed to the moon, and in consequence the method has been very rarely jtractised at sea. The precise in- stant of the phenomenon is, however, not necessary in all cases ; it is enough that the observer is certain that at one instant he sees the star, and that at another he does not see it; because the whole resulting error in the time of observation in this case, and therefore .u the longitude itself, cannot exceed the time elapsed between two sights of the moon. 872. The M.T. at Greenwich, at which the moon and the star to be occulted are in conjunction in R. A., is set down in the Nautical Almanac, as also the parallels between which the pheno- menon is visible. As it would require a distinct calculation to learn beforehand approximately the time at which the phenomenon will take place, the observer may content himself with finding, from the long, by ace, the time at place of the conjunction ; he must tlien, at an early opportunity, single out the star, and watch the progress of the moon towards it. In general, when the star is to the eastward of the observer at the time of conjunction, the phenomenon occurs beforfi that time ; when to the westward, it occurs afterwards. 1. Occultation of a Star. 873. TTif! Observation. Note the instant of immersion or emer- bion as nearly as possible. 874. The Computation. (1.) Find the Green. Date, and reduce to it the Sid. Time at mean noon, the moon's declination, hor. par-, iumI semid. ; reduce the hor. par. by Table 41. FINDING TllK LONGlTtDE. 32S [2.) Find the geocentric latitude by subtracting from the lai. the reduction of lat., Table f)2. From the time at place find the star's hour-angle, No. Gil. (3.) For arc A. To the prop. log. of the reduced hor. par. add the log. cosec. of the geocentric lat. and the log. sec of the star's dccl.: the sum is the prop. log. of arc A. For arc B. To the prop. log. of the red. hor. par. add the log. sec. of the gcoc. lat., the log. cosec. of the star's decl., and the log. sec. of the liour-angle ; the sum is the prop. log. of arc B. For arc C. Add together the prop. log. of the red. hor. par., the log. sec. of the geoe. hit., and the log. cosec. of the hour-angle; (louhle the sum, add to it the const. 1-582, and the log. cot. of the btar's decl. : the sum is the prop. log. of arc C. (4.) When the lat. and decl. are of the same name, add A to the fctar's decl. ; when oi contrary names, subtract it. When the star's hour-angle is less than 6\ subtract B from tiie star's decl. ; when greater than G*", add it Subtract C from A. Call the result the prepared declination. (5.) For Part I. of the ]> 's Parallax in R. A. Take the diff. lietncen the tnoon's decl. and the prepared decl.; under this ditf jiut the semid. : take the diff. and sum. Add together the log. cos. of the prepared decl., the const. 11761, half the prop. logs, of the diff. and sum : the sum is the prop. log. of Part I. For Part II. Add together the log. cos. of the prepared decl., tlie const. 1'1761, and the sum of the 3 logs, used in arc C: the Sum is the prop. log. of Part II. When the moon is on or near the meridian, this Part disappears. (6.) Apply Parts I. and II. to the star's R. A., thus : — Part I. In an immersion, subt^'act ; in an emersion, add. Part II. When the > is to the E. of the Mer., subtract ; when U'., add. The result is the moon's R. A. (7.) Find the G.M.T., as directed. No. 867 (6.) + 4 56 •© (Decl. S. lat. S. a.ld.) De.1. 5 16 11 -6 Ek. Dec 9th, 1825, lat. 9" 40 fi.. long, by ace. j itrved the immersion of x Aquarii,* W. of the aneridian ; Ur. Date, 9th 9'' 19"' 2-,' K.-.I.S.T.atm.n. 17 11 13-7 Ml". 7 19 57 Sur'i R.A. 22 2« 39 Hour-angle i 2 31 ' Arc A. H.P.;4'38"p- log. 0-5178 Geoc. Lat. cosec. 0-7777 + Decl. sec. coo 17 P. log. 1-2972 A, +9' 4"-S fl& C- 4 » -8 r 51' -W., at 7» 19" 57- required the longitude. 5 S. Re<l. Eq. U. P. Red. H. P. Semid. Lat. 9° 4 (Tab. 52) Cor. - P. log. 0-5178 o"oo6i 1-0487 0-0653 ■•6579 (Hotir-angle lex$ that 6*' gubtrart.) 9 36 19 • Thi» occultation, kindly furnished me hy the H i UvijiK been observed by biin, at tea, iu II. M. fr- ArcC. 0-5178 o-oo6i lour-angle cosec. 0-2928 '0-8167 c-S.e;- 2- 1-6334 Const. 1-5820 Hf DerJ cot. 1-0469 C,-o"-3p.log. 4-2623 I (^.7311 Capt. F. De Hoa, R.N. laate Creole. 324 NAUTICAL ASTROXO.MT. Prep. Uuc 5 Decl. Semid. Parti. 5° /43"-6 • 5 '» ?9 -6 cos. 9-99«2 5 i6 22 -6 const. 11761 3 43 '+ 53 '4 11 10-4 i pro. log. 0-6035 18 36 -4 ^ pro, log. 0-4928 -0 57 -9 pro. log. 2-2706 '^Hublraet, being immec.) Jan. 7tli, 1836, Bedford, lat. 52 1 ' Cos. of P . Const. Sum of 3 Ft. II. 1'" 1 * R.A. 1 J R.A. 1 -^19'' 1 At loi' G.M.T. Ship M.T. Long, in time ^S'28"N.,lon Part ■ep. De.l. logs. Arc C. + ."■ 50- -0 57- II 9"99»> 11701 2 p. log. 1-9910 DiflF. Sum. 22 30 45 19 30- 9 3 • 477> ', o'35"-6 2482Q 1 I 49 -4 8 -0060 3 p. log. ""^ Kx.2. 9 19 30 7 .9 57 ' 59 33 g.acc.-W. 3 3 or 29^53''9"W'. at loK 45... 53-2 M.T. obBeived the immersion of: Leonis, E. of the meridian : required the Longitude. Gr. Date io''47"', Red. S.T. 19I' 6'" 8-5, star's R.A. 10" 23'" 26-4, decl. 14° 58' 38''-8 N., J> icd.decl. i5°49'4o"N., H P. 55' 54"-9, Semid. 15' i6'-i, geocen. lat. 51° 57' 19". Arc A. 42' 33", D. 3'2i"-5, C. 2"-3. Prep. decl. i5"37'4S"o. Part I. 39'-9, Pt. 11. 2'" i2''-6. J R.A. loi' 20"' 33"-9. At lo'', lo'' 18"' ss'^S; at ii*', 10" 20'" 58*-5. ti .M.T. 10'' 48'" o'. By corr. of Part I. lo'' 47'" 45'. 2. Occultation of a Planet, 875. The Observation. The planet having sensible seniidiaineter, the plienonienon does not take place instantaneously. Note the instant of final disappearance, or the instant of reappearance. 876. The Computation. Subtract the planet's horiz. parallax from the reduced horiz. parallax of the moon. Also subtract its »emidianieter from the moon's semidiameter. In other respects proceed as for a star. 877. Degree of Dependance. A small error of Gr. Date will not sensibly affect the moon's parallax or semidiameter, and the de- clination is the only element liable to sensible error; Part I., there fore, is alone affected. To find the error in the long, in time, caused by 1™ error of Gr. Date. Find the change of decl. in l", add it to the diff. of declin., and recompute Part I. : the diff. between the result and Part I., as computed before, is the diff. or error of R.A. The error of long, in time will be, on the average, 30 times greater.* If the star pass very near the moon's upper or lower limb, the observation is not good. The inequality of the moon's surface, and an imperfect estimation ftf the figure of the earth, may cause small inaccuracies. The cases least liable to error on the several accounts enumerated are those which occur when the moon is near the meridian, and in which the central zone of the moon passes over the star. The emer- «ion from the dark limb is the case most distinctly marked. No. 862 applies to this observation. • Hence, to obtain the long, in time tiiie to !• or 15", the parallax In R.A. diu»I ba IfiK to 0*-003. This remark shews the ditticulty of ubtainiog eilreme precision troin anj •iuflc obiterTation. FTISDINO TFIE lONOITtDJt. Sf?5 V. By Eclipses of Jipitkr's Satellites. 8/8. Tiie eclipse or disappearance of a satellite in the shadow of the planet, called the Immersion, or the reappearance after eclipse, called Emasion, being a plienomenon which takes place at the same nhsolute point of time wherever the spectator may be pl.iced, affords ii ready method of finding the longitude. The diagrams of the positions of the planet and its satellites, as seen in N. lat., and other necessary information, are given in t!ic Nautical Almanac. The figures must be reversed in S. lat. It will be convenient for the observer to bear in mind, that when Jupiter comes to the meridian before niidniglit, the whole eclipse (both immersion and emersion) takes place on the E. side of the planet ; when after midnight, on the W. side. In an inverting telescope this will appear to be reversed. 879. Tke Observation. The telescope should have a magnifying power of not less than 40, and the observer should be ready some minutes before the time of observation, estimated by applying the long, by ace. to the time iti the Nautical Almanac. The sun should not be less than 8° below the horizon, nor Ju- piter less than 8* above it, for the phenomenon to be distinctly visible. 880. Tlie Computation. The difl^erence between the M. T. at place, found by observation, and that at Greenwich, is the long. Ex. Oct. 6th, 1822. near Igloolik, Int. 69° 21' N., immersion of the 1st satellite, lo'' 29'" 53 , M.T. The M.T. at Gr., in the Nautical Almanac, is \i,*' 56" o- ; the diff., jh jgm 27", long. W. 881. Degrf£ of Dependance. This method, tliougii easy and convenient, is not very accurate ; the eclipse is not instantaneous ; and the clearness of the air, and the power employed, affect consi- derably the time of the phenomenon. Observers have been found to (liti'er 40' or 50' in the same eclipse. The observation may be considered complete only when the im- mersion and emersion of the same satellite are observed on the same evening, and as nearly as possible under the same circumstances. Thus, if the satellite disappear a little sooner th;in if the air hail been clearer, it will emerge a little later from the same cause, and the mean of the two results iray l)c near the truth. The first satellite is preferable to tho others on account of the (frreter rapidity of its motion. 326 WACTIfAl, ASTRONOMY. CHAPTER VIII. Finding the Variation of the Compass. I. By the Amplitude. II. 13y the Azimuth. III. By Astro- nomical Bearings. IV. By Terrestrial Bearings. 882. The Variation is found by comparing the bearing of the sun or other celestial body, as shewn by the compass, with the true bearing as found by calculation. See No. 907. 883. When the time is known, the body may be observed, in the iiuiplest cases, at its passage of the meridian, at which time it bears due N. or S., or at its passage of the prime vertical, when it bears due E. or W. In other cases, the true azimuth may be found by calculation. When the time is not given the azimuth may be determined by observation of tlie altitude. When the altitude is nothing, or the body is on the horizon, as at rising or setting, it is usual to refer the bearing to the piime vertical, the angular distance from which (or the complement of the azimuth) is called the amplitude. The azi- muth may also sometimes be determined from the observed difference of altitude in a measured interval of time. The following rules are arranged more particularly for observa- tions of the sun ; but, after the explanations and precepts already given, no difficulty will occur in adapting them, when nece.ssary. to observations of other celestial bodies. I. By the Amplitude. 884. This method, which i* ))articularly convenient, is available twice a-day in fine weatiier, and at all seasons of the year. 885. The Observation* At sunrise, when the upper limb ap- is on the Iiorizon, observe its bearing, and continue to take rings of the centre, bisecting the sun's disc by keeping the up- k; * The usual instructions for taking an amplitude direct the snn to be ohsen-ed when his lower limb is half way between the centre and the horizon, at which time he is really on the hiiilzon, Nc. 433. But as it is not easy to seize the bearing at the required instant, and still less so to obseive several bearings e(|ually distributed on both sides of the proper position, whijb is essential to a correct result, the sun is commonly observed a whole diameter too low. The observation as recommended above is more convenient ir. practice, and the enxir »ri«ing from not observing the sun at the instant to wh>ch the true amplitude cor (No. 446 (1)), is removed by the correction. FINDING THE VARIATION OF THE rOMVASS 32? riKhl Ttirc on the upper liuib, until the lower limb ai>pear9. Read off each bearing?. At sunset, when the lower limb touches the horiz.on, proceed in like manner, until the apper limb disappears. See No. 2-2 1 . _ „, . . The mean of the readings, reckoning from the E. or M . poin'. i") the observed amplitude. 88b\ The Computation, by Inspection (1.) Enter Table 69 with t!ie Lat. and Declin., take out the amplitude, anrl mark it of the fiamc name as the Declin. (2.) Take from Table 59 A tlie correction. If this does not Hmount to nearly 1", it may in general be omitted. At liising. In N. lat. ajiply the corr. to the rii/lit of the ob- served amplitude. In S. lat. apply it to the left. At Setting. In N. lat. ajiply the corr. to the left of the ob- served amplitude. In S. lat. ai)ply it to the right. (3.) When the observed and true amplitudes are both N. or both S., their difference is the Variation. If one is N. and the other S., llieir sum is the Variation. Then, the observer being in the centre of the compass, when the observed am|>iitnde is to the left of the true, tlie Variation is East; when to the right, it is West. Ex. I. Jnue loth, lat. 17° N., long. IV W., olwerved sun's amplitude at setting, \V. ^° N. : required the Variation. Lat. 17", Decl. 23", Amp. W. 24° N. Obs. W.j£ N. Var. 16 W. Ex. 2. June loth, lat. 36° 40' S., long. 17'' W., obtained sun's amplitude at setting, W. ii''-3 N. : required the Variation. Lat. 36='7, Decl. 2 3°o, Amp.W. 29°-2 N. 37° and 13°, Corr. c" Obs. Amp. W. u El. 3. May 2gth, Ut. 47" N., long. 1 8- W., observed tlie sun's amplitude a» rising, E. 10° N. Lat. 47°, Decl. 21 J', Amp. E. ii"-; N. «o°and 22", Corr. o°o | p _ .. m Ob«.Amp.E. .o-oN.j E.^ N. Vab. 13 4W. Ex. 4. Sept. 25th, lat. f N., long. 151° E., observed the sun's amplitude at rising, E. 4° N. : required the Variation. Lat. 7", Decl i', Amp. E. :° S. Ob». Amp. K. 4^ N- Vab. s E. The Corr. hare ii o. The correction in Table .59 .\ i*> the same for a star or a |)larict as for the .sun, and is applied in the same way. When the moon is employed, the correction, which, in the case of the sun or a star, involves the sum of the dip and horizontal refraction, is the excess of her horizontal parallax over this sum. As the moon's hor. par. is 1°. anrl tlie refraction i°, in round numbers, this excess is about \°. which is nearly the quantify employed in Table ■')9A. 'i'his cor- rection, therefore, serves for the moon, but it must be ap|ili(Ml the contrary way to that directed for the sun. 887. The Computation, Accurately. (1.) Find the Greenwich Date and reduce the declination to it. (2.) To the log. sec. of the lat. add the log. sine of the declin.: the Huni is the log. sine of the amplituile. Apply the correction as Rborc. 328 NAUTICAL ASTRONOMY. 888. Bcgne of Dependance. In low latitudes the amplitucle ig (useeptible "of much precision ; in high latitudes refraction renders the result less certain. The relative temperature of the sea and ihe air produces no effect on the observed amplitude. IT, Cv THE Azimuth. 1. By Azimuth on the Meridian, 800. The Observation. When the sun approaches the meridian observe the azimuth, and continue observing till the same time after noon. The mean of the readings is the observed azimuth. When the sun is observed to the southward, if the observed bearing is to the E. of S., the variation is E. ; if to the W., it is W. When he is observed to the North, the contrary in each case. 2. By Azimuth from the Short Double Altitude, 891. The true azimuth is obtained from the observation of the dhort double altitude, p. 256, without regard to the apparent time. Case T. Observations on the game side of the meridian, No. 729. 892. The Observation. Observe the sun's azimuth during the interval between observing the alts., so as to obtain it at the middle of the interval. See No. 221. 893. The Computation. Having corrected the alts, and taken their difference, No. 729 (1), add together the log. sine of the diff. of a,lts., the log. cosec. of the interval,* and the log. sec. of the hit.: the Slim is the log. sine of the azimuth at the middle time from noon, nearly. El. (Ex. I, p. 358.) Lat. 34°40' S, diff. ofalts. 59' I, interval20'" IS', D. All. o" 59'-| 6in. 8 2^i;3 Int. 20'" 12" eosec. I 0554 Lat. 34° 40' see. 00S49 AziMiiiH 13°:^ sin. 9 3756 This azimuth cotnparcil with that observid woulJ afford the v.iriatioo. ♦ When it it intended to fina tne \'ariation by thii mi'thod at the same lime as tha l.atiiude, it will be convenient to take the sum of these three logs, first. The tiye logt. tniplo^ej in Ko. ''29 will tluis uft'ord two distinct results. FINDING THE VARIATION OF THE COMPASS. 829 894. Degree of Dependance. By adding to the result tlie difi". fur 30" ill the sine oF the D. alt., the effect on the azimuth of ^ in the diflF. alts, is seen, and the effect of an error, or small variation nf the D. alts, estimated. See also No. 679. Case TI. Observations on different sides of the meridian, No. 7:51. 895. The Obgervnfion. Observe the sun's azimuth when at the alt. nearest noon. See No. 221. 89fi. The Compufniion. Having found the time from noon of tlie greater alt., to the log. sine of this time add the log. cos. of the declin., and the log. sec. of the greater alt.; the sum is the log. sine of the azimuth at the time of observing the greater alt. Ex. (Ex. , p. 259.) Time from noon, I "•59- decl. 5.^°, greater alt. 49° 41 Great alt. 49" 41' Bin. 8-718 eos. 9-998 sec. 0-1 S9 Azimuth 4i° Bin. 8905 8. By Azimuth from Equal Altitudes. 897. The true azimuth may be obtained directly from the obser- vation of equal altitudes at sea, for time. No. 798. The azimuth, being computed as directed in No. 801, and compared with that observed at one or both of the times of equal altitudes, determines the variation. The altitude is required with more precision than for finding the time by the method, No. 798. This method is, however, not always eligible, because in low latitudes, where the observation of equal altitudes is favourable for the determination of time, the altitudes near noon are great, and therefore unfavourable for the observation of the azimuth. See No. 889. 4. By Azimuth on the Prime Verticnl. 898. The Ol)serv(ifion. Having found by Table 29 cither tho app. time or the altitude at the instant of the passage of ihe prime vertical, begin to observe a little before that time, and continue observing till the same time afterwards. The mean of the readings, when it is not accurately E. or W., is the variation. A.M. If the sun bear to the northward of E., the variation is E. ; if to the southward, it is W. P.M. If the sun bear to the northward of W., the variation is W. ; if to the southward, it is E. 899. As a celestial body, when on the prime vertical, changes its azimuth more slowly than at any other time, an error in the appa- rent time will be of little consequence, and the method will be found one of the most convenient in practice in high latitudes during the BIX months that include the summer. 330 NAOTICAL ASTRONOJrr. 5. Bi/ Azimuth deduced from an AKttvde, 900. TJie Observation. Take bearings of the sun's centre, notiiig the time of each reading. Take an alt. as soon as convenient before and after the bearings, noting the times. 901. The Computation. (1.) Having found the mean of the azimuths and of the corresponding times, reduce the alts, to the mean of the times, No. GCO, reduce the decl., correct the alt., and find the azimuth, No. 678 or 674. Ex. Feb. 19th, i828,r.ji.,Paia Bay, Nnples, lat. 40"'5o' N., long. I4'3' E., Mr.Fislipr observed the mean ot seven azimutlis of ihe snn by Kaier's compass. N. 223° 24' E. (ur 8. 43° 24' W.) Sun's true alt. 33" 34' j sun's reduced decl. n" 14' S. By Expl. Tab. 5, lat.4i°,alt.33\''M 15S9 N 57 5 Oecl. Ilj°,dist. 159 Dep. 310 (lesser) Sum 885 Dist. 100 and D. Lat. 88 5 give cnuife or Az. S. 28" W. Ditto observed 43^ Tab. I5i \V. 6. By Azimuth deduced from the Time, 902. The observation is already' described in No. 900. (1.) Find the Green. Date, to which reduce the declination and the elements employed in finding the hour-angle. (2.) Compute the azimuth, No. 675.* Ex. I. June 23rd, 1829. P.H .at Constii.tinople, lat. 41" l' N., long. 28° 59' E. ; tbe mean of seven times by cliroii. 4'' 43'° 15". and nf seven a/.imiitlis of the sun, observed by Mr Fisber with Kater's i.ompa>s, botrteen 286" 30' and 2SS', was N. 287° 16' E., or N. 73' 44' W- Reduced pol. dist. 66° 33'. Time 4" Cliron fast on A.T. 8un*« Hour-angle 4 Pol Dist. 66" 33' ColU. 48 59 43" > 5* 32 43 half 2" 19- 51* 17 34 »ec. o 27297 cos. 999488 tan. 0^2316 co«e«. 007269 sin. 9 183 83 14° 28' tan. 9 41183 half 57" 46' 8 47 69' 19' 14 28 ArJmnth N 83 47 W. Do. observed N 72 44 \V. Vab. ni~3 \V. Ei. 2. Dec. 27tb, !83I, Lisbon, lal. 38" 42' N., Ion?;. 9" S' \V., Mr. Fisber nh.crv.d »be mean of ten azimuths of the sun by Kater'.s compass (lietwein 165" and 166° 50') to be N 166" 7' E. The mean of the limes by chron. (between lO*" 7"' 30" and 10'' 15'" 45") nss 10'' II'" 47'. Chron. fust on A.T. 42"" l8"; red. pol. dist. 113" 22'. Computed Ai. N. 143° 44' E. ; Vah. 22' 23' W. • The work of finding the Azimuth is much lessoned by the use of suitjil'le tables. Burdwood and Davis's .\zimuth tables and Star Azimuth tables extend from the equator to 60" latitude, and are published in a convenient form riy .T. D. Potter, Hii Minories. London, E. Such tables are indispenEable for the navigation of iron shipa. See also Ltcky'e ' Wrinkles," for et»r». riNIUNG TIIK VAUIATION OF THE COMPASS. Slil III. By Astkonomical T5karings. 903. The true bearing of a point of land, or other terrestrial object, may be determined by means of the difference of hearing between it and the sun, or other celestial body ; the true bearing of the latter being deduced by observation, or computed from the time. The difference of bearing may be obtained directly by observing with the compass the bearings of both the sun and the object ; or by the sextant, when the sun is on the horizon. But as the obser- vation of two bearings at the same iustant cannot always be con- veniently made, the angular distance between the sun and the object is measured by a sextant or circle, and the bearing of the object alone observed. The difference of bearing is then deduced, by calculation, from the observed angular distance and the alti- tudes of the sun and the object. The true azimuth of the object being thus obtained, the varia- tion is deduced. 904. The Observation. Observe the sun's alt., then the angles between the objectand thenearestand farthest limbs; lastly, observe the sun's alt., noting the times of each contact. Take the alt. of the object, at the point from which the sun's distance is measured. When the variation is required at the same time, the bearing of the object must be obtained as nearly as possible at the time of the observation of the angular distance. 905. The Compntation. (1.) Find the means of the times and angular distances, and reduce the sun's alt. to the mean of the times. Find the Green. Date, and reduce the sun's decl.; find his pol. dist., correct the obs. ang. dist., and the alt. of the object for index-error, when necessary. .\ole For common purposes, when the observer is not much elevated and the alt. r^ the object docs not exceed a few minutes, the sun's decl. may be corrected at sight, tha dip. refiaction, paral'.a*, and ihe alt. of the object neglected, and the precepts (2) and (4) «iniitied. (2.) FiiTd the app. alt. of the sun's centre (by applying the ind.- Torr. dip, and semid.), and thence the true alt. by subtracting the refr. or corr. of alt. (3.) Find the sun's true azimuth. When the sun is not near the meridian, this is found by No. 674. When he is near the meridian it is better found from the time, No. 675. The lat. will be required more correctly as the sun is nearer tiie meridian, nnd less so as he is farther from it. (4.) For the corr. of ang. dist. arising from the point observed not being exactly on the true horizon. Take the diff. between th? ob.a. alt, of the object and the apparent dip, Table 30. 5b2 NAUTICAL ASTRONOJIT. To the log. sine of the reiiiiiiiuler add the log. sine of the sun'g Bpp. alt. and the log. cosec. of the ang. dist.: the sum is the log. sine ot the correction of the ang. dist. When the dip is lexs than the alt. of the object, add the corr. to the ang. dist. ; when the dip is the greater of the two, subtract it. (5.) For the diff. of azimuth. To the log. cos. of the corrected ang. dist. add the log. sec. of the sun's app. alt. ; the sum is the log. COS. of the diff. of azim. between the sun and the object. When the ang. dist. exceeds 90°, take the supplement of the arc found as the diff. of azim. (6.) For the Variation. Apply the diff. of azim. to the sun's azim., according to the case, which will be best understood by draw- ing a figure : the result is the true azim. or bearing of the object. The true bearing compared with that observed shews the variation. Kt. Dec. 4th, 1819. at 7'' 30" A.M., in Pernamhuco Koa/i, lat. S" 4' S., loiiR. 3^° S^' ^^'■> M. Givrv took the following alts, and angular dist., height of the eye 16 feet, ind.-corr. o — {Man.surl'Emploi,^c.) 7 26 10 23 23 1 190° 30' 30' Alt. 10 Mean 23 '9 5 95 '5 '5 Corr. 0. ( Green. Date, .3d, 21-47" Ued.Dpcl. 22' 10' 47" S. I'd. D.St. 67 49 (2) Olis. Alt. Arp. Alt. 23' '9'8 + 12-2 True Alt. 23 30 COS. 9-97788 ,. >q. 9 83071 (4.) Corr. of Ang. Disi sine Mlf) 10'- (dip) 4' --6 0) Alt. 23' 30' l)if.t. 95 >5 4 2 Corr. Ang. Dist. 95 I J 7242 ine 9601 osec. 0002 (.S & 6.) Computation of Diff. of Azim. Ang. Dist. 95° 17' COS. 89642 O Alt. 23 30 6CC. 00376 Suppl. O Az. Ohj. Az. Observed Vak. 84" '4 95 46 S 69 15 E. .S 26 3J \V, S 31 40 w. ~5 "W. FINDING THE VARIATION OF THE COMPASS. S'i'^ IV. By Terrestrial Bearings. 906. The true bearing or aziimith of a mountain, at a consider- able distance, is determined from its geographical position and that of the observer. As the true azimuth and the course on tiie great circle are the same thing, the problem is that in No. 339 ( 1 ), p. 133. But as mountains are rarely seen much beyond a hundrnd miles, it is near enough to proceed thus : — Find the D. Lat. and D. Long, between the places in minutes of ai-c. Turn the D. Long, into Dep., No. 318 or 319. Find the Course. No. 28U (1). This is the appro.'cimate azimuth. With the mid. lat. as a course, and the D. Long, as dist., find the Dep. ; this is a number of minutes, one-half of which is to be subtracted from the approx. azim. ; the remainder is the true azimuth, very nearly. ¥.x. Lat. 5o°6'N., long. 142" 50' W., find the true azim. of Mt. St. Elias in lit. 60° 18'. long. 140° 52'. D. Lat 12 and D. Long. iiS give Dip. 586, and Course 78° 26'. Then 60° and 118 %ivc Dep. loa 2 ; and 51' subtracted from 78° 26' gives the A/.iM. N. 77° 35' E. In low I»litiides, and in all cases when the object is near N. or S., the correction may be neglected. (For more predsion, see No. 39.5, p. 151.) 907. The term Variation, as defined in No. 882, and used in this chapter, is the difference between the true bearing of any object and its bearing by a compass. From what has been said in Chapter IL, this quantity must differ from the coiTect varia- tion by the instrumental error of the compass, by the local effects of the land, and, further, on board ship, by the deviation. There may be instrumental errors in a compass, which cannot be detected unless the correct magnetic bearing of some object ia known. For this reason it is desirable, when there is any reason to suspect the accuracy of the standai-d compass, that advantage should be taken of being in a port where the exact variation is known, to examine the compass according to the process described in No. 224. Errors in observed bearing, arising from the sight- vane not being vertical, or from the reflector being out of place, may be avoided by using low azimuth's amplitudes, or nearly horizontal bearings of terrestrial objects. Errors arising from tlie centre of the card not being in the same vertical plane as the line of sight, may be avoided by taking bearings of several objects distributed round the horizon. The true bearing of one object may be determined by process III. or IV., the others by horizontal angles therefrom. The effects of such local disturbances as are mentioned in No. 222 may generally be eliminated, either on land or at sea, by oLseiving iu several positions, with the view of getting ou oppo- 3.34 NAUTICAL ASTRON'OirV. Bite sides of the disturbing cause, and taking the mean of tlio results as the correct variation. When an observation is made at sea with a compass which ia instrumentally correct, and is free from local disturbance of the land or ground, the difference between a true bearing and a com- pass bearing, commonly called the Total Error, enahies the navigator to shape a correct true course. This is in general all that is actu- ally required for navigation. But such an observation would not dotermine the variation, unless the deviation is exactly known. A good value of the deviation may be obtained by interpolation, if the ship has been swung a short time previously, and again a short time after. Allowing the same ou the total eiTor will give the variation. When the compass is well placed, the mean of the total errors on two opposite cardinal points is a good value of the variation. A still better value may be obtained by taking the mean of the total errors on the four cardinal points. To obtain an accurate compass bearing, it is necessary that the ship's head should be steadied as directed in No. 248. When a ship's head is moving to port or starboard, the compass card is obviously liable to be dragged round in the same direction as the head is moving, by the friction on the pivot. On the other hand, in iron ships it has been found, that when the head is moving to the right, the compass-needle stands a little to the left of its due position, and vice versa. The last mentioned effect of the ship's mo- tion in azimuth is especially noticeable when the ship's head is near the north or south points. It is due, possibly, to the transient magnetism not instantly adapting itself to the position of tiie ship, as she moves round in azimuth. An exact bearing can be obtained by taking the mean of two, taken with the ship's head moving in opposite directions ; also an accurate deviation-table may be quickly obtained by turning a ship round to port and to star- board under steam, making use of the sun's azimuth, and taking the mean on the four cardinal points a« the variation, where it is aot otherwise known. Reduction of the True Course to the Course by Compass. 908. When the true course to be steered is determined, it must be reduced to the course by compass. The variation of the compass is to be applied (No. 221) ; the result is the correct magnetic caurtie. See p. 1.59. When the total error (No. 907) of the compass is known, it is to be applied to the true course, otherwise the deviation (No. 227) must be applied to the correct magnetic course; the result is the course by compass. CHAPTER IX. TiiK Tides. I. PUKNOMENA OF THE TlDfcU IF, RULES FOR FIND'NG THE TlMK OF Hkjh Watek. III. Tide-Observations. this chapter we shall attempt merely a general enumeration of the principal phenomena of the tides, with such other matters as aia In this the pri; of direct practical importance I. Phenomena of the Tides. 909. Tiie connexion observed in all ages, and, with particular exceptions, in all places, between the succession of high waters and the moon's meridian passage, has established the belief that thu moon is the cause of the tides. The principle of gravitation,ton which the motions of the eartli and the celestial bodies are calcu- lated, and their figures explained, has confirmed, and at the same time corrected, tliis belief, by shewing that sensible ett'ects must be f)roduced not only by the moon, but also by the sun, tiiough, from ler greater neaiiiess, the moon has by far the greater influence; and the general result would, naturally, until the observations were analysed, be attributed exclusively to her. 910. The attraction of the moon acting most strongly on those parts of the ocean which are nearest to her, that is, over which she is vertical, tends to draw these parts towards her, while their place is supplied by the water at the sides of the globe. And since the central parts are likewise more affected in the same action than the surface at the opposite or farthest side, the figure of the earth becomes elongated in the direction of a line drawn towards the luoon ; that is. the water is accumulated at the point exactly under • The readier may refer, tor ftddilional information, to various papers, by Sir John Lul>- l,ook and the Rev. Dr. Wliewell, in the Philosophical Tranaactiona, Jkc, 1833, particularly t.i " An EsBay towards a Map of Cotidal Lines," followed by other dissertations by l)r Wliewell ; and to " The Tides," by Professor George Howard Darwin {John Murray, Albemnrrle Street). t Thii^ princijjle is that there eub^iists ainnngst all particles o{ matter a mutual attrai tiua «hoM indei.fiity la inveraely as the square of the distance. ?,r,6 NAUTirAL ASTRONOMY. the moon, and at nnotlier point distant from tlie former 180° in latitude and longitude. The moon, in her progress to the westward, causes thus, at each meridian in succession, a high water, not bv •Irawing after her tiie water first raised, but by raising continually that under her at the time. The opposite high water, or, as it is called, the inferior tide, would, if tlie moon's action was uninterrupted, follow the other, or superior tide, after the interval of half a lunar day, or 12*' 24"' on the average. Again, the sun, acting in the same manner, though with less force than the moon (in consequence of his distance more than counterbalancing his greater magnitude), produces two tides, which would follow each other, if uninterrupted, after an interval of half a solar day, or 12 hours. 911. But, instead of four separate tides produced by the inde- pendent actions of both bodies on the mass of waters in their orig'u d ibrm, the effect produced is the same as if, after one of the bodies, as the moon for example, has given a form to the waters, the sun alters that form, the two separate actions thus producing a joint result. Hence the place at which it is high water is that at which the sum of the heights of the tides produced by the two bodies is greater than any where else. 912. When the sun and moon are on the meridian togetlier, their actions concur, and the tide is higher than at any other time. Tlie same holds when they are in opposition. These highest tides are called spring-tides, and occur after new and full moon. Again, when the sun and moon are 90° apart, their actions tend to neu- tralise each other; and the 7ieap-tides, which occur after the first and third cpiarters of the moon, are the smallest of all. (See No. 919.) 913. Since the sun and moon act with greater force as they are nearer, the effect of each body in raising the tide is gieater as its parallax is greater (No. 436). The highest spring- tides would occur, therefore, in January, about the time of the month when the moon's hor. par. is greatest. But the effect of both bodies is greater, generally speaking, as their alts, are greater, since when vertical the effect is greatest. This yieriod, therefore, depends on circumstances. 914. If the actions of the sun and moon were, as we have hitherto supposed, uninterrupted by obstacles or forces of any other kinds, the tides would be regular, and their calculation certain. But from the unequal depth of the ocean, and the barriers presented by continents which stand across the natural progress of the tides, their motion is interrupted, and the tide-ioave (as the accumulation of waters is called), abandoned by the forces which originated it, becomes subjected to the mechanical action proper to waves in general. 913. It is necessary to distinguish between the motion of a wave and that of a current. A wave is not an absolute transfer of the body of moving water in the direction of the motion of the waves, Lut is a motion i>i;rppndicular to the surface, or up and down. The TnKTIDKS. .''>o7 u.otidii of waves is roprcjiMiteil Iti tlio tlutlcriiig of a Hat; and llie -liakinLf of a sail. It is easy to see that liiis i;iii(l of motion is roiiipatiblc with iinmeiise velocity, v.itlioiit any ai>|)refiable current in tlie water itself; tlir.s the tide-wave appears to pass from tiie Cape of Good Hope to Cape Bianco in twelve hours. 916. The motion of waves is quicker as the water is deeper. Also, the largest waves are the swiftest; a fact illustrated by tlus superior velocity of a heavy sea over that of the rippling of a pool. W hen tiie water slioals, the wave is retaided and becomes steeper on the advancing- side, as is seen in the approach of waves to a shelving shore, and in the bores of rivers. The v^ilocity of waves is also considered to be greater as tiieir length (or distance from hollow to hollow) is greater ; thus the tide-wave, though inferior in height <o the waves of an agitated sea, yet travels with prodigiously greater velocity. Waves of diHerent size and velocity merge into one an- other, as is known to those who have endeavoured to follow with the eye the waves of the sea. Lastly, when the waves meet wilh tilistacles, such as sand-banks or reefs, the directions of their motions, as well as their figures, are changed. Several of the anomalies which the tides present are attributed to these and like circumstances.* 917. The current wliich accompanies the tide, and changes its direction with the ebb and How, is the effect of the alteration of the level of the water during the passage of the tide-wave. Also, when a body of water in a channel has been set in motion, the motion does not immediately cease with the cause that produced it. Hence the tide-current does not necessarily, and in all cases, change with the tide; and thus, under certain ciicumslances, the current of the ebb continues to run for some hours after the flood-tide has made. It is considered probable that many of the anomalies in recorded times of tide have arisen from thus confounding the time of high oi low water with the time of slack water. Admiral Btechey, who bestowed much attention upon the com- plicated movements of the tides on our Western coasts, states that though each point of the coast in the Irish Channel has its propel time of high water, yet the turn of the stream takes place simul t;ineously to all, namely, about the time of high water at Mon-coinbe Hay. This time is nearly that of Liverpool; accordingly, in order to know whether the stream is setting into the Irish Channel or out of it, It is necessary merely to find whether the tide is rising or falling at tliis place, 'fhus while the tide-wave, in coming in, is making it high »attr at the different places succeeding each other in its progress, the stream is, nevertheless, running ont.|- * Among tlie mo«t curious of these effects are. those called mier/ertncet, whereby two iJi.iiiict «fts (>f waves may, in their combination, produce aiiiiurent irst. See H/iil. Trant. 1-5?., p. '54. Or this pr